Get yourself a merino wool base layer shirt, like a thin one not sweater thick. It'll be surprisingly cool.

Try hemp or bamboo fibre (has different names like rayon, tencel, lyocell).

You could try different thicknesses of cotton and see how it feels. Or cotton blended with the other materials. Some companies list their shirts as lightweight or heavyweight or give cloth weight in gsm (grams per square meter) or oz (ounces per square yard). A lightweight and oversized shirt will be good in the heat.