I think wide-brimmed hats are very good "covers" where you aren't moving through the trees and brush. They keep the direct rays of the sun from burning your face, ears and neck. Critters that might drop down on you will probably not get into your hair and the sun protection is very good from mid-morning to mid-afternoon.
Some prefer a baseball-type hat with a rear flap but that leaves your ears exposed and part of your neck.
I think where you are going to be traveling may dictate if a baseball hat, a GI boonie hat or something along the lines of a stylish Akubra shown above would be what your need. I like the boonie (ugly but efficient) because you can cram it into anything and it looks as bad as it did before you treated it that way...
For around camp. maybe a plain-old baseball hat but if I was going to be doing a lot of hiking where shade from trees was not available, then the full brimmed hat style would be good.
I think the baseball style hats with the "beavertail" soft neck protection are great. Nothing like a sunburned neck and ears to ruin a night's sleep!