I find that my canvas Tilley, which I dearly love, makes my fore head sweat a lot. The hat band isn't particularly absorbent. Sweat builds up and then drips down into my eyes and onto my sunglasses. Not good.
I switched to an el cheapo tourist cotton hat that I bought in Maui. It is cooler, lighter, and soaked up the water better, but it's the ugliest hat.
I recently bought a nylon Tilley that is cooler, lighter, and far quicker drying than my canvas Tilley. It seems to be doing the trick.
Just thought I'd share my experience in hopes that it might benefit someone.