If you have that perfectly distressed cotton tee that you love, wear it. Several studies have shown that there’s little thermoregulatory benefit to wearing synthetic shirts over cotton, though synthetic shirts might be more comfortable.
As for a cotton/polyester hat, you can wear that, too. One study showed that donning a cap won’t make your core temperature hotter than going without—but it can feel hotter.
Into compression? Even in higher temperatures, you can wear your compression socks. They may lower your perceived rate of exertion without significantly raising your core temperature.