While at other times: conjunctive adverb?

Summary

This is largely a question of style. Not all punctuation (or absence thereof) is obligatory. There is a continuum of styles concerning non-obligatory punctuation, from a light style that favors little punctuation to a heavy style that favors lots of it (CGEL, 1727).

In your sentence, there are eight conceivable combinations of comma placement. Of these, only two seem to never be used; see [1] below. Of the rest, the most common practice would put just a single comma before while; the second most common practice would put no commas at all. But the remaining four possibilities are seen as well. In particular, the influential Chicago Manual of Style would arguably recommend enclosing while at other times by a pair of commas; see below.

The semicolon before while is most often used when the dependent clause (after while) is of significant complexity, usually with further commas. Your sentence arguably doesn't qualify on that basis. However, if you wish to put a special emphasis on the dependent clause, it is arguably fine to use a semicolon. See "CMoS's recommendations concerning the semicolon" near the end of this answer.

As far as grammatical status, CGEL would say that while is a preposition—though one that 'falls at the boundary between relative words and prepositions'—which takes a content clause as complement (p. 1267).

You were right when you said that at other times is an 'adverbial phrase'; that is indeed a very common terminology. CGEL, however, would say that it is a preposition phrase (PP) which functions as an adjunct of time location (p. 696).

A disclaimer/apology

The detailed discussion of these issues will, unfortunately, take some space. There are many different aspects to this question. It is not surprising that it causes much confusion for everyone!

Discussion

Observed patterns of comma placement in English corpora

As you pointed out, in your sentence, there are three places where commas could possibly go. In principle, this produces eight possibilities: 2 × 2 × 2, since each comma can either be present or not. I'll use a three-digit binary number to code which of the three commas are present, according to the scheme (1st digit) while (2nd digit) at other times (3rd digit).

Two of these possibilities, (010) and (110), are definitely not acceptable:

[1]  i  ^*He may often be considered gregarious and social while, at other times he is more reserved. (010)
      ii  ^*He may often be considered gregarious and social, while, at other times he is more reserved. (110)

More precisely, I don't think I have ever seen them in published work written by educated native speakers. In particular, I haven't seen them in my searches (explained below) of the three English corpora.

That leaves the following six possibilities. They are ordered from the lightest to the heaviest in the used style of punctuation. Actually, ii (001) and iii (100) are a tie; however, v (011) is heavier than iv (101), because the commas are closer together in v than in iv.

[2]  i  He may often be considered gregarious and social while at other times he is more reserved.  (000)
      ii  He may often be considered gregarious and social while at other times, he is more reserved.  (001)
     iii  He may often be considered gregarious and social, while at other times he is more reserved.  (100)
     iv  He may often be considered gregarious and social, while at other times, he is more reserved.  (101)
      v  He may often be considered gregarious and social while, at other times, he is more reserved.   (011)
     vi  He may often be considered gregarious and social, while, at other times, he is more reserved.  (111)

If you search for 'while at other times' at the Corpus of Contemporary American English (COCA), the British National Corpus (BNC), and google books, you will see that the most common pattern is (100), followed by (000), and then (101) as a distant third. Here are some examples:

Listeners and their interests are understood to often benefit from intervention by the state or civil society, while at other times they benefit from deregulation.  (100)

At times the bass is a harmonic instrument while at other times it creates lines, sounds, and rhythms that drift freely  (000)

Sometimes the debris was twisted metal, while at other times, the rubble revealed some kind of machine.  (101)

The remaining three—(111), (011), and (001)—are rarer still, but one does encounter them:

When related to the individual, the concept has at times had a very inclusive meaning, encompassing interest in honor, glory, self-respect, and even afterlife, while, at other times, it became wholly confined to the drive for economic advantage.  (111)

At times they admire Jesus and grieve over his plight while, at other times, they demand his death.  (011)

…at times was executed by the household servant under the direction of the conversa while at other times, the mistress of the house did the task herself.  (001)

As far as a possible semicolon instead of a comma in front of while, the same three corpora show that it is sometimes used when what follows (or even when what precedes) while is particularly complex, usually containing further commas. Here are two examples:

That is, no one ever explains to me why sometimes I shouldn't repeat some things I say; while at other times, some other things I say would not only be all right, but would be considered so funny they would be repeated so many times for so many people's enjoyment.
^{From COCA}

For example, sometimes it lifts us out of a culture of unduly promoting one's family, mutual interest group, or “old boys” network at the expense of others; while, at other times, it immerses us more deeply in personal, collaborative relationships.
^{From David J. Kettle, Western Culture in Gospel Context Towards the Conversion of the West: Theological Bearings for Mission and Spirituality (2011) (link)}

Having said that, the following are also in COCA. But to my eyes, they are a bit jarring, especially the first one:

The relation was linear for Co; while Zn and Se showed respective non-linear relations.

Two taught days add breadth and expertise to learning; while the trainers, all clinicians in practice, have first-hand experience in the field of study.

Light and heavy punctuation; optional delimiting commas

The main thing to keep in mind is that many times, commas are optional and a matter of style. In particular, CGEL (p. 1727) speaks of

a distinction between light and heavy punctuation styles that is independent of regional and publishing house variation:⁵

[5]  i  On Sundays they like to have a picnic lunch in the park if it's fine.         [light]
      ii  On Sundays, they like to have a picnic lunch in the park, if it's fine.    [heavy]

This distinction has to do with optional punctuation, especially commas: a light style puts in relatively few commas (or other marks) in those places where they are optional rather than obligatory.

^{5Alternative terms are 'open' and 'closed' respectively.}

The type of punctuation that is relevant to your question is delimiting commas, which (p. 1745)

mark both left and right boundaries of a subclausal constituent that is set apart from the main part of the sentence, usually indicating that it is in some sense less central to the message. If the left or right boundary coincides with that of a larger construction that is marked by a stronger indicator, then the comma is superseded by, absorbed into, the latter.

In your sentence, at other times is an adjunct of temporal location (see below). As it happens (p. 1746),

With adjuncts, there is considerable variation as to when delimiting commas are used: this is the area where the contrast between the heavy and light styles of punctuation is most evident.

The main factors influencing the use of delimiting punctuation are:

[27]  i  length and complexity of the constituent
         ii  whether or not there are punctuation marks nearby
        iii  the linear position of the constituent
        iv  the semantic category of an adjunct
         v  the possibility of misparsing
        vi  prosody

So things can get complicated, especially when there are more than one of these factors to consider. Just for the sake of illustration, here is part of what CGEL says about the first three (p. 1746):

Other things being equal, a short simple constituent is less likely to be marked off than along complex one (e.g. one with the form of, or containing, a subordinate clause). The influence of nearby punctuation is seen in such a pair as:

[28]  i  She was not sorry he sat by her, but in fact was flattered.
         ii  She was not sorry he sat by her but, in fact, was flattered.

In [i] we have a comma before but, separating the coordinate main clauses, and the following adjunct in fact is not marked off. Conversely, in [ii] there is no comma before the coordinator and the adjunct is delimited. It would be possible to combine the comma of [i] with those of [ii], but to have three commas in such close proximity is likely to be perceived as noticeably heavy punctuation.

As for position, delimiting commas are most likely with adjuncts located internally within the clause. And they are more likely with elements in front position than at the end of the clause.

The grammatical structure of your sentence

According to CGEL (p. 1267),

when and whenever, while and whilst fall at the boundary between relative words and prepositions taking content clause complements;

Moreover (p. 737),

the primary meaning of while and whilst is durational, but they have a secondary sense equivalent to whereas, as in

While/Whilst the first act was excellent, the second seemed rather dull.

The meaning expressed here is contrast, not co-duration.

In your sentence (I will use the most common pattern of comma placement),

[3] He may often be considered gregarious and social, while at other times he is more reserved.

it is clear that while has the secondary, contrastive meaning: you could replace it by whereas without any change in the meaning of the sentence. Whereas is also a preposition, one of those that govern non-expandable content clauses (CGEL, p. 971). Here 'non-expandable' means that the content clause cannot be preceded by that.

So, in your sentence, at other times he is more reserved is a content clause. Note that it is of a type that could stand as an independent clause (I will use a light style of punctuation):

[4] At other times he is more reserved.

Here at other times is a preposition phrase (PP) functioning as an adjunct of temporal location (CGEL, p. 696).

One observation that will be of interest below is the following: while in your sentence could be replaced by but without any change in meaning (although, to my ear at least, the while version does sound better):

[5]  i  He may often be considered gregarious and social, while at other times he is more reserved.
        ii  He may often be considered gregarious and social, but at other times he is more reserved.

In other words, although while is not a coordinating conjunction, its function is very close to one. I will use this argument several times.

Delimiting commas in your sentence

Putting it all together, there are no obligatory commas in your sentence. There are three optional ones, as discussed above.

Relevant recommendations of The Chicago Manual of Style

Since I keep saying that the comma placement in your sentence is largely a matter of style, it makes sense to look at recommendations of at least one prominent style manual. A particularly influential one is The Chicago Manual of Style (CMoS).

As far as comma placement with dependent clauses, the relevant recommendation in CMoS is this:

6.25: Commas with dependent clauses following the main clause

A dependent clause that follows a main, independent clause should not be preceded by a comma if it is restrictive—that is, essential to fully understanding the meaning of the main clause (see also 6.27 [which further discusses the distinction between restrictive and nonrestrictive clauses, especially in as it regards that vs. which]). For instance, in the first example below, it is not necessarily true that “we will agree to the proposal”; the dependent if clause adds essential information.

We will agree to the proposal if you accept our conditions.
Paul sighed when he heard the news.
He wasn’t running because he was afraid; he was running because he was late.

If the dependent clause is merely supplementary or parenthetical (i.e., nonrestrictive, or not essential to the meaning of the main clause), it should be preceded by a comma. Such distinctions are occasionally tenuous. In the fourth example below, the meaning—and whether the subject is running or not—depends almost entirely on the presence of the comma (compare with the third example above). If in doubt, rephrase.

I’d like the tom yum, if you don’t mind.
At last she arrived, when the food was cold.
She has a point, whether you agree with her or not.
He wasn’t running, because he was afraid of the dark.

or

Because he was afraid of the dark, he wasn’t running.

Arguably, the interpretation of the main clause in your sentence (he may often be considered gregarious and social) does not depend at all on the information in the dependent clause (while at other times he is more reserved). Thus, it would seem that CMoS would recommend a comma before while.

Having said that, remember that we noted that while in your sentence could be replaced by but without any change in meaning. In other words, although while is not a coordinating conjunction, its function in your sentence is very close to one. For now, note that this type of analogy would also result in CMoS recommending a comma before while:

6.22: Commas with independent clauses joined by coordinating conjunctions

When independent clauses are joined by and, but, or, so, yet, or any other coordinating conjunction, a comma usually precedes the conjunction. If the clauses are very short and closely connected, the comma may be omitted (as in the last two examples) unless the clauses are part of a series. These recommendations apply equally to imperative sentences, in which the subject (you) is omitted but understood (as in the fifth and last examples). (For the use of a semicolon between independent clauses, see 6.56.)

We activated the alarm, but the intruder was already inside.
All watches display the time, and some of them do so accurately.
Do we want to foster creativity, or are we interested only in our intellectual property?
The bus never came, so we took a taxi.
Wait for me at the bottom of the hill on Buffalo Street, or walk up to Eddy Street and meet me next to the Yield sign.
Donald cooked, Sally poured the wine, and Maddie and Cammie offered hors d’oeuvres.

but

Electra played the guitar and Tambora sang.
Raise your right hand and repeat after me.

As for the adverbial phrase,

6.31: Commas with adverbial phrases

Although an introductory adverbial phrase can usually be followed by a comma, it need not be unless misreading is likely. Shorter adverbial phrases are less likely to merit a comma than longer ones.

On the other hand, his vices could be considered virtues.
With three consecutive swings, Jackson made history.
In 1931 Henrietta turned fifty.

but

Before eating, the members of the committee met in the assembly room.
To Anthony, Blake remained an enigma.

Also, while it is about independent clauses, this may nevertheless be of interest:

6.32: Commas with a participial or adverbial phrase plus a conjunction

When a participial or adverbial phrase immediately follows a coordinating conjunction, the use of commas depends on whether the conjunction joins two independent sentences. If the conjunction is simply a part of the predicate or joins a compound predicate, the first comma follows the conjunction (see also 6.23).

We were extremely tired and, in light of our binge the night before, anxious to go home.
The Packers trailed at halftime but, buoyed by Rodgers’s arm, stormed back to win.

If the conjunction joins two independent clauses, however, the comma precedes the conjunction (see also 6.22).

We were elated, but realizing that the day was almost over, we decided to go to bed.

Strictly speaking, it would not be wrong to add a second comma after but in the last example. Such usage, which would extend the logic of commas in pairs (see 6.17), may be preferred in certain cases for emphasis or clarity. See also 6.26.

Again, while is not a conjunction. But we've noticed above that in your sentence, it could be replaced by one (but) with no change in meaning. By that logic, it would seem that CMoS actually recommends the (101) pattern (He may often be considered gregarious and social, while at other times, he is more reserved.)

CMoS's recommendations concerning the semicolon

Again, while is not a conjunction, and your sentence is not composed of two independent clauses. But it is very close to being that way, because while can be replaced by but with no change in meaning. Thus, the following recommendation may be relevant.

6.59: Semicolons before a conjunction

Normally, an independent clause introduced by a coordinating conjunction is preceded by a comma (see 6.22). In formal prose, a semicolon may be used instead—either to effect a stronger separation between clauses or when the second independent clause has internal punctuation. Another option is to use a period instead of a semicolon; see 5.203.

Frobisher had always assured his grandson that the house would be his; yet there was no provision for this bequest in his will.
Garrett had insisted on remixing the track; but the engineer’s demands for overtime pay, together with the band’s reluctance, persuaded him to accept the original mix.

or

Garrett had insisted on remixing the track. But the engineer’s demands …

One disclaimer, however: in the case of while, for all its similarity in function with but (at least in the sentences we are considering), I do not think it would be acceptable to put a period in front of while so that the dependent clause is a separate sentence. In particular, I don't think a period would work even in the cases where a semicolon does work. Using the examples I quoted above,

That is, no one ever explains to me why sometimes I shouldn't repeat some things I say. ^*While at other times, some other things I say would not only be all right, but would be considered so funny they would be repeated so many times for so many people's enjoyment.

For example, sometimes it lifts us out of a culture of unduly promoting one's family, mutual interest group, or “old boys” network at the expense of others. ^*While, at other times, it immerses us more deeply in personal, collaborative relationships.

As I mentioned, the semicolon seems to work best with while when the dependent clause is particularly complex. In the following recommendation, we see a similar logic at work (and much the same ground is covered in CGEL on p. 1740). This strengthens the case for claiming that the most common situation in which a semicolon is used to separate the main clause from a while-headed dependent clause is one in which the dependent clause has a particularly complex structure with internal commas.

6.60: Semicolons in a complex series

When items in a series themselves contain internal punctuation, separating the items with semicolons can aid clarity. If ambiguity seems unlikely, commas may be used instead (see 6.19). See also 6.129. Note that when a sentence continues beyond a series (as in the third example), no additional semicolon is required.

The membership of the international commission was as follows: France, 4; Germany, 5; Great Britain, 1; Italy, 3; United States, 7.
The defendant, in an attempt to mitigate his sentence, pleaded that he had recently, on doctor’s orders, gone off his medications; that his car—which, incidentally, he had won in the late 1970s on Let’s Make a Deal—had spontaneously caught fire; and that he had not eaten for several days.
Marilynn, Sunita, and Jared, research assistants; Carlos, programming consultant; and Carol, audiovisual editor, provided support and prepared these materials for publication.

but

She decided to buy three watches—an atomic watch for travel within the United States, a solar-powered, water-resistant quartz for international travel, and an expensive self-winding model for special occasions.

Having said all that, I will admit that in the corpora I see quite a few cases where the semicolon seems to be used because the main clause, not the dependent clause, is complex. This practice cannot be explained by pointing out that while functions similarly to but in these sentences and that therefore people were applying recommendations meant for but—as we just saw, all such recommendations concern the cases where it is the clause that comes after the conjunction that is complex. Perhaps such examples are better explained by the author's perceived need to put additional stress on a comparatively 'light' dependent clause after a very 'heavy' independent one.