bald men know hats
The baseball hat is likely the US favorite these days, but I only wear one around town. The baseball hat was developed,... you know,... for baseball. The long narrow brim shades the eyes on a fly ball, and while batting, .....but that's about all it does well. Afield, I find they have few attributes.
I'm not a cowboy either, so though I own a western style hat, I don't wear it much, the wide brim gets in the way of eye relief on scopes, (for me anyhow), effects my anchor point with my bow, and the hat is awful if there is much brush about.
My field, stream, hunting hats all have a 2-3" brim all they way around, to shade my eyes, forehead, ears and neck. That is important UV protection from skin cancer later. A baseball cap will not give you that. That same brim will keep ran and snow away to a degree. For over 30 years, in cooler weather afield, I have worn a brown felt trilby/fedora style hat made by the Dobbs company. Yes, one hat, thirty years. The liner and headband are gone, the wool felt crown and brim is stained but still functional. The trilby and fedora are two distinct styles, but this old hat seems to combine both the low crown of the fedora, with a modest brim more like a trilby. Think something like Sean Connery or Bear Bryant wore, but brown. I'll add that the modest brim can be tucked up under a hood on a parka, something a cowboy hat will not allow.
In hot weather, I've got several variations of the US boonie/jungle hat. The woodland camo version works fine for warmer spring gobbler hunting. The full brim breaks up the shape of the human head too. (so I think). I have some tan/khaki ones that seem not as hot in the sun. I have another made from light weight nylon sort of stuff that is likely the coolest. Some have snaps on the side to pull the brim up Aussie style if I want. In REALLY hot weather, I have a true sand colored staw fedora with about a 3" brim.
The old Dobbs was expensive, even in the late 80's. The boonies are available everywhere for $10-15 bucks. The straw/sand fedora was about a $30 dollar hat, but I only wear it as a dress hat.