As a plein air painter, finding the right equipment is essential. Getting out on site and finding out your easel is rickety or your palette isn't big enough is a shortcut to disaster.
You'll understand, then, why finding just the right hat is paramount.
Keeping the sun off the brain pan is as important as remembering to bring your paints along. There's keeping the head cool as well. Sweat trickling down into the eyes does not facilitate clear vision.
The garden variety straw hat will suffice if need be. Inexpensive and light, they're usually wide enough to keep the sun off the neck and shoulders, and loosely woven enough to let air flow through. The down side is that they also have a tendency to catch gusts of wind, and it's a bit distracting to be choked by your own hat while it attempts to set sail still attached to your head. Straw hats are also fairly brittle and do not hold up well to being sat upon, which you will do sooner or later.