"Under/straight from the horse's mouth" — etymology?

You're taking Mr Philby far too literally. He is referring to the 'high ranking official' as 'the horse's mouth.' This is a very common English allusion to the idiom "straight from the horse's mouth."

The second person works directly for 'high ranking official' and so is under 'the horse's mouth' as any functionary is under their boss. Thus, in Philby's mind, his acquaintance might know much of what 'high-ranking official' knows.

There is no such idiom as "under the horse's mouth." It's just Philby's phrase.

The original meaning of the expression

There is some dispute about the underlying sense of the phrase "straight from the horse's mouth." John Ayto, The Oxford Dictionary of English Idioms, Third Edition (2009) agrees with Stan Gipple's assertion (above) that the term comes from horse racing:

(straight) from the horse's mouth from the person directly concerned or another authoritative source. This expression refers to the presumed ideal source for a racing tip and hence of other useful information.

To similar effect, The Wordsworth Dictionary of Idioms (1993) offers this note on "straight from the horse's mouth":

Possibly originally used for 'infallible' tips in horse-racing.

But several other reference works argue that the phrase "straight from the horse's mouth" refers not to reliable insider information metaphorically spoken by a horse (such as how confident it is that it will win an impending race), but to the mute testimony of the horse's teeth. From Martin Manser, The Facts on File Dictionary of Cliches, Second Edition (2006):

from the horse's mouth, straight From the best authority. The analogy here is to examining a horse's teeth, which reveal its age with some accuracy. Although this fact has been known for centuries (and indeed gave rise to the adage, DON'T LOOK A GIFT HORSE IN THE MOUTH, dating to the fifth century), the expression dates only from the 1920s. "I have it straight from the mouth of a horse," wrote Christopher Morley (Kitty Foyle, 1939).

From Christine Ammer, The American Heritage Dictionary of Idioms (1997):

from the horse's mouth From a reliable source, on the best authority. {Example omitted.] Also put as straight from the horse's mouth, this expression alludes to examining a horse's teeth to determine its age and hence worth. [1920s]

From Nigel Rees, A Word in Your Shell-like (2004):

(to get something) straight from the horse's mouth To hear something directly from the person concerned and not garbled by an intermediary. The horse itself is not doing any speaking, oof course. A horse's age can be judged best by looking at its teeth (which grow according to a strict system). So, if you are buying a horse, you do better to look at its teeth than rely on any information about its that the vendor might give you. Known by 1928. 'Meanwhile, it was a privilege. Straight from the horse's mouth into the note-book. The boys scribbled like mad'—Aldous Huxley, Brave New World, Chap. 1 (1932). The Horse's Mouth was the title of a novel by Joyce Cary (1944).

And from Robert Hendrickson, The Facts on File Encyclopedia of Word and Phrase Origins, Second Edition (1997):

straight from the horse's mouth. By examining a horse's teeth, an expert can make a good estimation of its age; a horse's first permanent teeth, for example, don't appear until it is about two and a half years old. So, despite what any crooked horse trader might have wished them to believe, informed horsemen in England stood little chance of being cheated about a horse's age—they had it on good authority, straight from the horse's mouth. The expression came into racetrack use in about 1830 and was part of everyday speech by 1900.

Unfortunately, Hendrickson doesn't cite any concrete instances to corroborate his claim that people used the phrase at racetracks in about 1830 and in everyday speech by 1900. The expression does not appear in Jon Bee [Badcock], Sportsman’s Slang; a New Dictionary of Terms Used in the Affairs of the Turf, the Ring, the Chase, and the Cock-Pit (1825) yields no mention of "horse's mouth," which suggests that the expression was not current at that date.

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Early Google Books matches for the phrase

Confirming Rees's timeframe, the two earliest Google Books matches for "from the horse's mouth" are from 1928. The United States Catalog: Books in Print January 1, 1928, includes this entry:

From the horse's mouth. Mandel, G. ed. $3.35 McBride, R.M. co (1956-)

And St. George's Gazette, volume 46 has this on page 21:

Suffice it to say that the said horse must be a knowing creature, and fond of "special." This version is straight from the horse's mouth. Serjt. Wood proved that there are other methods of guiding a horse than by the reins—to wit, gripping the saddle firmly with the hands front and rear. That his hands were skinned is neither here nor there.

And on page 130:

Coming events therefore are:—Camp, on 19th August; before that, Dublin Horse Show for some; and after that, move to York, in October (latest rumour, straight from the horse's mouth).

The odd thing here is that the one of the matches from 1928 appears to be from the United States (McBride is a U.S. publisher) and the other from Britain, so these results don't provide any definite insight into where the wording originated.

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Update (January 19, 2021): Earlier matches for the phrase from Australia

A search of the Elephind newspaper database turns up fifteen unique matches for the phrase from the period 1910–1927—before the earliest Google Books match. The most striking thing about these matches is that all of them come from Australia. Indeed the earliest U.S. instance of "straight from the horse's mouth" doesn't appear until October 20, 1939, in comments by the U.S. ambassador to Japan cited in "U.S. Envoy's Outspoken Criticism Shocks Japan," in the San Bernardino [California] Sun.

Here are the first five unique matches from Australian newspapers. From "Picking Them," in the [Adelaide, South Australia] Daily Herald (May 12, 1910):

Many hunters have many ways of using the divining rod of fortune (says an exchange). The pure gambler picks them Chinese fashion, which means the letters of the alphabet are compared with the initial letters of the names of horses in a race until a corresponding initial is found. Some allegedly more enlightened sports obtain their griffin "straight from the horse's mouth," per the second cousin of the wife's brother of the trainer, jockey, stable boy, or somebody equally "in the know" connected with the stable.

Pretty clearly, this instance involves the horse (through its familiars) talking about its condition and confidence for the coming race.

The next instance uses the expression metaphorically and so whimsically that its sense is difficult to interpret. Nevertheless, it appears in six newspapers in Victoria and New South Wales during May 1916. From "Old Noah's Almanac: The Only Reliable Prophet in War Time," in the West Gippsland [Victoria] Gazette (May 2, 1916):

Friday is always an anxious day, Suppose something should go wrong in the night, and the week-end, Saturday and Sunday, didn't turn up! Then to-morrow would be Monday gain, and no pay-day. A horrible thought.

But this week it is almost a one-horse snip. Old Noah has had a few words with Jupiter and Canister, the Dog Star, and can assure his customers that all will be well. There will be a to-morrow, and it will be Saturday. We have this straight from the horse's mouth.

The next instance is even more widespread in Australian newspapers—fifteen altogether, again mostly in Victoria. From F.W. Thomas, "Sam the Sprucer: He Tells of a Talk With a Pessimist," in the Swan Hill [Victoria] Guardian (May 17, 1917):

"'The Army in France!' I says. 'You take it from me, mister, that's all swank,' I says. 'There ain't no Army in France, and never was. All this talk about millions of men joining the Army is all flapdoodle. I know for a dead cert, because I've got n aunt that used to be in service with a man whose uncle knew the brother of a man whose lodger's sister's husband's wife was married to the bloke what used to take the tickets on the Boulogne boats, and he says that there hasn't been a single soldier go to France since the war broke out. That's straight from the horse's mouth, that is! If he don't know, nobody don't.

This item is noteworthy because the character "Sam the Sprucer" is presented as a British soldier, suggesting that F.W. Thomas is a British—not Australian—writer.

From an advertisement for Balranald Follies in the [Balranald, New South Wales] Riverina Recorder (December 8, 1920):

Full of DAINTINESS and PEP. Full of MELODY and MIRTH. A tip straight from the horse's mouth:—"Book at once. Plan rapidly filling."

And from "Stage Gossip: Lines of Laughter," in the [Sydney, New South Wales] Sunday Times (February 12, 1922):

Harry C. Musgrave has added another link of laughter to the most chucklesome chain of comedians that could possibly girdle a continent. The new link is Malcolm Scott, an English humorous singer of note, who is one of the big outstanding figures in English vaudeville. Mr. Scott wears skirts, but they are not the skirts of Aladdin's mamma, or the pantomime ladies who carry on washing as a profession. Far from it. Scott's array is of the finest for he represents personages of no less distinction than Queen Elizabeth, and the equally late Queen Ann. Hi's language—straight from the horse's mouth, as it were—may not be historically correct, but London theatre-goers have found it deucedly funny.

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Even earlier matches for the phrase from British newspapers

A search of the British newspaper Archive finds instances of "straight from the horse's mouth" from as early as 1861. The earliest of these is from Bell's Life in London and Sporting Chronicle (September 22, 1861), in what appears to be an advertisement for a racetrack tout sheet of some kind, although the OCR snippet that BNA provides is so wretched that the meaning and context are difficult to discern (and I don't have a subscription to this service):

Rank Outsiiler. A raker to jn straight from the horse’s mouth, and two steamers fop places. Ganihridgeahlre a dead heat. Honor delay. Send an addressed envelope, and eight stamps teovanls wise# Hiroct Silas Wcrrlfleld, stud groom.

An equally nonsensical snippet is drawn from Bell's Life in London and Sporting Chronicle about six years later (July 20, 1867):

BITTER ALMONDS, and Southampton, and all meetings near London. FREE FROM PRUSSIC ACID. tj * j j This delicious essence may safely used for flavouring custards, STRAIGHT from the Horse’s Mouth. —Goodwood blancmanges. Ac. and all kinds of pastry. brakes. The pots boil over every one. The winner is not quoted. Sold retail chemists ...

A much more coherent snippet appears an item drawn from the Manchester Courier and Lancashire General Advertiser (July 30, 1883) [combined snippets]:

... was committed [by] a pot-house politician, who advanced to the female player and without the least provocation thrust the brass instrument into her mouth with such force that she had to desist playing. I would have these molestors of public order know that it more creditable to be [a] street musician than one of the tribe loafing about the above quarter, and living on the gullibility of those foolhardy enough part with money for a tip straight from the horse's mouth. The frequent occurrences of conduct similar to the above makes one feel the [lack] of safety whilst shouldering against this rabble, whose clothes [are] more creditable than their manners. I hope our authorities will not be remiss in reminding policemen on the several beats where these corner men congregate to keep vigilant eye upon their behaviour.

From the [London] Clarion (December 29, 1894) we have a figurative use of the expression, applied to a different sporting context:

Only last week, I was told by a gentleman who holds an important post in Lancashire football, that wages were stopped a fortnight since by the officials of one of the Competition clubs. This, the gentleman informed me, was straight from the horse's mouth, and the information was not in connection with any of the suspended clubs. He would not volunteer any name.

And from the [Lancashire] Athletic News (May 29, 1899) [combined snippets], in the context of an impending horse race:

...suburb of London, and that with a little self-sacrifice they may be enabled to witness — win the Derby. We have information straight from the horse’s mouth [as] to which will win, but do not wish to create jealousy amongst certain members of our staff by giving it, although we might go so far as to say that the quadruped to pull it off will need Fly, and not be guilty of any Foxing to get out of the Hclocanste and the ways of the Oppressor.

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Conclusions

The expression "straight from the horse's mouth" seems to have originated in Britain, by the late 1860s, probably in the context of inside information about the condition of an entrant in an impending horse race. The expression migrated to Australia by 1910 but seems not to have become common in the United States until several decades later.

On balance, I think it is more likely that the original sense of the expression is a fanciful or jocular one in which a horse tells how it feels prior to a big race, rather than a more serious one of a horse giving mute testimony about its age simply by the length and condition of its teeth.