Discussion Forum—A Way with Words, a fun radio show and podcast about language

The Phrase Finder says:

It would be nice to be able to report that this graphic phrase alludes to some event where someone used a hat to disguise their speech, or similar. The only practice that comes close to fitting the bill is that of 'topping', in the UK parliament. To be allowed to make a 'a point of order', which is an interruption to a previous speech in order to query something that had been raised, MPs had to be 'seated and covered'. That is, seated and wearing a hat. Topping was the name given to 'talking out' a bill, otherwise known as filibustering, by continuing to speak until debating time ran out. Having made a point of order and while wearing a top hat, an MP couldn't be interrupted and could continue talking for as long as he/she wished. Naturally, as these speeches often lasted hours, they were frequently filled with rambling nonsense.

On the other hand, he asserts

This began life in the USA, in the late 19th century, with a slightly different meaning from the present one. It then meant to bluster.

Thr Grammarphobia Blog says

The Random House Historical Dictionary of American Slang and the Oxford English Dictionary cite this quotation from the New York World in 1888 as the earliest printed example: “Dis is only a bluff dey’re makin’ – see! Dey’re talkin’ tru deir hats.”

But a quick stroll through the Ngrams Viewer shows Edward Irving said in "Breakers Ahead!: An Answer to the Question, Where are We At?", that

The silver men said that the Senator was "talking through his hat," and that the result of unconditional repeal would be a further fall in prices.

This was in a discussion of the Sherman Anti-Trust act, which was passed in 1890, so the book was probably not published in 1804, as Google asserts, and although there are a number if cites from the 1890s, nothing before the 1888 cite.

One observer speculated that the phrase might go back to Joseph Smith translating the Book of Mormon, where he put the stone on his hat, and holding it up, offered an oral translation.  There's absolutely no evidence offered in support of that theory, but none opposing it, either.

If we're going to speculate - fun, and entirely worthless - I'm more likely believe that holding your hat in front of your face allows for deniability.  "Somebody said, I don't know who" would work well if those repeating the statements want to shield you without actually lying, yet introduce a controversial statement into the discussion.

So to answer your precise question - does anyone know - I think the answer is "probably not", but I doubt it came from Parliament.  It most likely came from the western US.