Mexican Sombreros along with similar styled Spanish and Mexican hats I believe. Forage hats were also popular. Most hats were passed down the generations so there was a lot of variation. Though bowler hats used to be a lot more horse riding friendly in their Wild West era days, so the more successful likely would have worn them while riding.

Crazy fact as well, the bent up look for modern cowboy hats mainly came from the fact that age and wear of older hats meant their ends started turning up. This eventually became a style in its own right.

So a bit like holes in denim jeans became a trend through older jeans getting holes in them, bent up hats became a trend due to older hats seeing their ends bend with wear and tear.