That that's kind of cool, I'm not sure covering your face would work unless you were caught in a sand storm. I went out today and I took along a pair of long pants in case I needed them. It was hot, and I figured if I put some ice cubes in them and rolled them up, they would be cool and wet. I was climbing Sugarloaf Mountain, and I never needed them, but I used them another way. When I got to the top, I had sweat getting into my eyes, so I took them and unrolled them and cooled and cleaned my face. It worked very well, because once you start getting sweat in your eyes, it usually keeps getting worse, but that stopped it, and cooled me off too.
That gives me an idea, that perhaps you could take a large towel and use it for keeping yourself cool by wrapping ice cubes in it and putting it on your back or wrapping it on your head. I don't get overheated easily, so I'd probably just wrap some ice in a towel and throw it in my backpack for when I wanted to cool off some.