Even very fine hair can be hot if it’s curly like mine. When it gets long, it acts like a wool hat, with the trapped air insulating. Last year, I took Ye Olde Scissors to it before RAGBRAI, and am very glad I did! I made sure to put sunscreen on my ears, though, and a bandanna on the nape of my neck.
Much like your cap, a bandana run under cool water and then put around the neck is *wonderful* on a hot day!
I’ve been wearing a dry bandanna under my helmet lately, on days when it’s warm enough to not need my new merino wool balaclava (I now understand why people rave about merino!). The bandanna works well — confortable, and thin enough to not affect my helmet adjustment. Seems like one might work well for those of you whose “hair color” tends towards pink. . .!