A man and his hats...
As an avid outdoorsman, I try to always wear a hat with an all-around brim, for protection from the sun and rain. I almost always wear such a hat, except in the most severe cold, in which case I wear a hat that incorporates ear protection.
But rather than a "cowboy hat" per se, I tend to favor one variation or another of the "Tilley Hat" (I own quite a few, some from Tilley and some from Ultimate Hats, some in cotton and some in nylon).
Depending on how I configure the brim, it can look like a Fedora, a "boonie" hat, a "safari" hat, or even an "Aussie" hat. I wrote a little piece once (on another firearms site, from which I am now banned, but that's another story) about how I'd used the pocket inside the crown to conceal a little .380 semi-auto (when I had nowhere else to carry it)! That's the type of hat I wear whether in NYC or in Mexico, or points in between. I even wore one in England.
Because I live within an hour's drive of the Stetson Outlet Store (in St. Joseph, MO) I have a small collection of Stetson hats, although I don't wear them too often. As a sartorial purist I don't think it's quite fitting to wear a "cowboy hat" without also wearing "cowboy boots". But I'm very proud of the Stetson "Open Road" (aka "LBJ Model", in "silverbelly" of course!) that I acquired quite a few years ago at the Stetson store for a mere $10!
When I go to the range, I go "one better" than a cowboy hat; I often wear a campaign hat. It's not an el cheapo; the felt is so heavy and stiff that the crown almost functions as a helmet. But I've put a decided bend in the brim, so I like to think I look more like Richard Farnsworth in "The Grey Fox" than like Jack Webb in "The D.I."
But you'll never see me in anything resembling a baseball cap (let alone one worn backwards)!
Cheers, "Ringolevio"