Summer Scottish hat/cap

I've got a kilt, sporran, renaissance shirt, and I need a hat of some kind. My hair is thinning and it's important I have a head cover, but I don't want heatstroke from wearing a wool Tam. AI claims there are linen and cotton Tams, but I cannot find them. Doesn't have to be a tam. But I don't want to wear my Scotland baseball cap. Anyone have any suggestions, stores, links?