Amazon link, ugh, but I love these hats for running. Super lightweight, hold up in the washing machine (not dryer), good at keeping sun off my face and sweat off my eyes. They fit both my medium-sized woman head and my husband's big ole man head.
https://www.amazon.com/2XU-UR1188f-run-cap-black/dp/B002ZU8LCC/
2XU RunCap. 2oz
2XU run cap black https://www.amazon.com/dp/B002ZU8LCC/ref=cm_sw_r_other_apa_cogUybN1EHZ26