I agree that visors are terrible if you are interested in protecting your head from the sun. They only shade your face, you'll sweat simply because the top of your head is baking in direct sunlight.
+1 for Tilleys.
There are many brands that make great hats...Outdoor Research, Columbia, Patagonia, The North Face, REI (only available in REI stores), even Nike has decent baseball cap styles.
If you go to camping/fishing/sporting goods stores, you should find a wide variety of quality hats (and visors if you're so inclined) to try on. Whatever you choose, I still recommend you pick one up that has UV protection and breathable fabrics/vents. That way you not only protect your entire head, but you get good air flow, too.
A quality hat is a worthy investment.