I have an older Milan straw hat that I bought on ebay. The sweatband was dry rotted and left little brown speckles on my head. I took it out. I tried the hat on without a sweatband and you can hardly tell you have a hat on. It's very comfortable.
Does the sweatband have anything to do with the integrity of the shape? I imagine that since all hats have sweatbands there is a reason for this. Will I ruin this hat if I wear it and sweat in it without a band? Or, does it just help the keep the pug from becoming sweat stained?