Speaking of Texans, proper cowboys traditionally wore a range of hats, most commonly bowler hats*. The iconic Stetson wasn't invented till 1865 (and in its original incarnation looked more like an Amish hat) and didn't gain popularity till the very end of the 19th century.
*Billy the Kid favoured a top hat.
In the cowboys & injuns stories I read as a nipper they always wore 10-gallon hats. I just looked up the origin of this term and it appears to be a mistranslation thing. (They were originally 8-gallon hats, but that got metricated to 38-litre hats, and rounded up for the conversion to US gallons.)
Not sure what the difference is between a 10-gallon hat and a Stetson.
I do know, however, that in France a bowler is called a melon hat, as in Chapeau Melon Et Bottes De Cuir, which is what they call The Avengers (as in John Steed and Emma Peel RIP, not Iron Man and co).