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I have used a wicking neck gator for years with great success. Something like this
I put it over my head so it covers my neck and then just pull the back up to cover the top of my head. This keeps your helmet from making your head itchy, covers your neck from the sun, and wicks away any sweat. You can wet it for additional evaporative cooling. Trust me, it gets hot in Arizona!
We wear these, with under our motorcycle helmets and our ski helmets. One piece of advice, if I may, go with something other than grey or black. They tend to come off with your helmet, and you want something you'll spot easily in that parking lot.
https://www.racibabi.com/Helmet-Liners.aspx
Stop by your local bicycle shop. They may have what your looking for too.
I wear a liner sold by BMW called Ride Hat. I think it's silk.