Many hundreds more students

Yes, it is acceptable. Here are some attested examples of similar usage:

Many hundreds more rebels were dead now.
If there were any delay, many thousands more people would die.
The Germans had expended seventeen trainloads of ammunition and many thousands more men.
^{All from British National Corpus (BNC)}

…their own well-trained, well-armed ground troops over the many thousands more lives of the defenseless victims of these perpetrators.
^{From Corpus of Contemporary American English (COCA)}

According to CGEL (p. 356), many and more belong to the lexical category of determinatives, and are both degree determinatives.

As far as hundreds, this is a cardinal numeral. Since it is here inflected for grammatical number, it is actually a noun. As CGEL says (p. 385),

The cardinal numerals are primarily determinatives but they have a secondary use in which they inflect for number and hence belong in the noun category: They set off in threes/enrolled in their hundreds. In practice, only low or round numerals are used in this way.

Next, I would like to argue that many hundreds more is a constituent, i.e. that we have the following constituent structure:

[ [many hundreds more] students].

(I will deal with the internal structure of many hundreds more below.)

Constituency analysis is sometimes quite difficult and there can be pitfalls. I don't claim that the arguments I'm about to present are irrefutable, but I hope that they will suffice unless and until someone presents a positive argument that this is not the right constituent structure.

First, we have argument from coordination; in general, only constituents can be coordinated:

many hundreds more or many hundreds fewer students

Yes, there are exceptions, and, without further analysis, argument from coordination is usually not sufficient.

But here is a second argument: the fragment answers test. Normally, only a constituent can answer a question, while retaining the meaning of the original sentence (see e.g. here):

Many hundreds more students arrived. -> Q: How many students arrived? A: Many hundreds more.

Again, these are not irrefutable arguments, but I hope they will do for now.

Assuming that many hundreds more is indeed a constituent, it would function as determinative phrase (DP). Here are some other examples of DPs (CGEL, pp. 23, 54, 385, 395, 424, 431, 432):

almost every/all         too many                                            some/any more
all the more                not every                                              hardly any
at least ten                  more/less/fewer than twenty         considerably/a lot more than fifty
no/a little more          greater than 20 000

As far as the constituent structure of the DP itself, it would seem to be

[ [many hundreds] more],

with more as the head of this DP.

The arguments would again be similar. First, from coordination:

many hundreds or even several thousands more

Second, from the fragment answers test:

Many hundreds more students arrived. -> Q: How many more students arrived? A: Many hundreds.

Once we accept that [ [many hundreds] more] is the constituent structure of the DP, we look for the head. This is a DP, so the head should be a determinative. This excludes hundreds, because, as we said, that is a noun. But many hundreds is clearly an NP, with hundreds as the head. This means that many cannot be the head of many hundreds more. Namely, a basic postulate of constituent structure is that it is hierarchical: the ultimate head of a phrase is also the head of all subconstituents of which that head is part. If many were the head of many hundreds more, the hierarchical postulate would demand that it also be the head of the subconstituent many hundreds. But we have just seen that many is not the head of many hundreds; thus, it cannot be the head of many hundreds more. This leaves more as the only possibility for being the head of many hundreds more.

If all this is correct, the full constituent structure is

[ [ [many hundreds] more ] students].