Out in the sun, like back in the day when I took pictures in the ring at horse shows, nothing beats a straw "cowboy" hat...Weston or the like.... They shade, maximize the cooling of evaporation, are light, and kept my fair skin from blistering.
Ambling around in WI, a ball cap works just fine. Woodswalking, boonie until the weather turns cold, then rag wool or heavy fleece knit cap to keep ears from freezing off are my choices.
Cold night camping, a light fleece head cover retains heat for the night.
Someday...maybe...I'll spend the dollars for a muskrat fur trapper's hat.
Can't consider that expense now, but boy are they warm!!!