Except in the cold (-20C), I wear an Outback Trading Kodiak hat. Waxed canvas and a wire rim.
I don't need to shoulder arms such that the barrel hits the brim. That's a wasted second when working fast moving game.
If it's windy, I'll add the rodeo strings and jam it on securely. Rarely an issue in our forests.
Just yesterday was looking at Akubra and Stetson for a seriously good hat.