Where does 'talking through your hat' come from?

The earliest use of 'talking through a hat' (and variants) that I found was a prosaic, straightforward simile in the 1864 novel Clara Vaughan, by R.D. Blackmore:

I hear a gruff voice as if it came through a hat.

The next use I found, in the 1879 Decoration Day, an oration at the Academy of Music, New York, Chauncy Depew credits his use of "talking through his hat" to "the slang phrase of my Bowery friends":

After reading all the speeches, some good, some indifferent, some bad and some incomprehensible, except on the theory best expressed by the slang phrase of my Bowery friends that the orator was "talking through his hat"....

So Depew's use of the phrase was to make sense of nonsense. That comports with the contemporary meaning:

to talk about something without understanding what you are talking about

Cambridge Dictionary

Depew's sense, however, may intimate what appears to have been the more common early meaning of the slang phrase, a sense which is perhaps closer to the literal. That meaning found expression in another early print use:

The very latest remark about the young man who has corralled everything from beer to champagne the nght before and gets around in the morning morose and silent is that "he is talking through his hat." It is said that this is quite as expressive as anything he could say.

The Sun, New York, NY, 11 Nov 1886 (fourth column, toward bottom).

As described, the slang phrase evokes the mumbling, fragmented speech of a drunk or person suffering from a severe hangover, speech that sounds much as might be heard from somebody actually talking through a hat.

In light of this early, more literal meaning, the earlier (at least as early as the 1820s) 'in his hat', a phrase said to be Irish in origin and signifying "drunk" (see also the British slang sense of 'elevated', that is, "intoxicated"), should be mentioned as at least a possible influence on the development of the contemporary meaning of 'talking through his hat'.

British uses of 'talking through his hat', which as far as can be determined (by me, who else) first appeared in the 1890s, seem to be sponsored by the American, as witnesses this from the Manchester Evening News (paywalled; Manchester, UK) of 20oct1891 (emphasis mine):

One wonders what would have been the effect upon the colonel's enthusiastic nature if he had had another bottle of wine, or upon the veracious chronicler of the important speech if the colonel had accompanied his utterances by a defiant nod. It doesn't seem to have occurred to the said chronicler that the colonel was "talking through his hat," as the Americans say.

In contrast or as supplement to the possible influence of the Irish 'in his hat', press accounts in the US from 1900 propose an even earlier origin (from the 17th century):

  Somebody has discovered that the slang "Talking through his hat" did not originate in America, but was first used by Moliere, the French dramatist, who in his "Miser" makes one of his characters say: "To whom I speak? I am speaking to the inside of my hat." — Philadelphia Times

Whatever the scope of the influence of the Irish 'in his hat' and the speech in Moliere's "Miser", the US origin of the phrase, and hence the British origin, is evidently from Bowery (New York) slang of the mid-to-late 19th century.