I'm a HUGE tester of warm weather gear and have been an official gear tester of some large outdoor clothing firms. I second what Vash says. By far the best sun hat I've ever tried was made by Sun Day Afternoons. I use the "Sport Hat", though their original Adventure hat is their most popular. As Vash points out, it covers the head entirely, has a Coolmax sweat band, is UPF 50+, has cooling vents, etc. It is THE sun protection hat to have IMO.
Sun Hats for Men
In second, I'd choose a hat by Outdoor Research. I've used their Helios Sun hat and their Sun Bucket. The Sun Bucket is one of my favorite hats. Great features, a good sun protection...though the nape of your neck might get sun because of the lack of a flap. Best thing about Outdoor Research is their warranty...LIFETIME. And they honor it quickly, too...with no questions asked.
Outdoor Research | Men's: Men's Hats, Warm Hats, Winter Hats, Sun Hats | Outdoor Research | Designed By Adventure | Outdoor Clothing & Gear
You can't go wrong with either of these companies offerings IMO.
FWIW, I've recently (within the last year) tried the greatest warm weather shirt technology I've ever tried. CooL Q Zero technology by Mountain HardWear. The Chiller Long Sleeve shirt (converts to short sleeve) is made with thousands of small, blue rings on it. When your sweat touches the rings, they puff up an actively keep you cool. It worked so well that a couple of times I was in the shade I had to move back into the sun. Yes..it works that well. Incredible and hard to believe. And stylish too! Nice enough to wear to dinner.