Never a fan of cycling caps under helmets until long distance rides in the sun. Now I bring two. Synthetic so they’ll wick better, evaporate for better cooling, irritate less.
Though I typically spray sunscreen on my head if I’m planning to stow the cap. (Also, a good cream like Neutrogena sunblock or Banana Boat Faces helps reduce sweat carrying the stuff into your eyes which can ruin a day)
I echo the advice from the ginger rider above: white arm covers are also huge, even on hot days. Taking a break from the sunscreen and rays on the arms some days cuts down on fatigue.