I would probably put my vote with the Tilley hat crowd for more extreme climes (like the deserts around my tiny corner of the world).

However, if you're willing to explore some less well-known or well-used headgear, you could always hide in the shade of the wide brim of a campaign hat (don't worry, this one doesn't come with the Boy Scouts pin). You can see one of Baden-Powell's kilted assistants wearing an identical hat here.