The bucket hat, which I think of as The Gilligan, is a popular choice for archers. Particularly noticeable in competitive archery, I guess it's light, it blocks the sun from bugging your eyes if you pull it down and it doesn't get in the way of your string. But as I consider different jazzy sorts of hats like cloches and tyrolers and newsboys, they're probably just as good. As long as you're not wearing a baseball cap or maybe a ten-gallon or Merlin-styley one, your headgear is probably A-OK. A short brim, or even a longer, flexible brim that'll mush back, probably won't interfere with your string as you pull it to your anchor point.
Also, what we really need on the blazingest days is a shuttle from the free public range to a free public pool, hey?