This straw hat purchased for protection from the sun and was frequently used when I was doing medieval reinactment on hot days.
The wide brim protects the back of neck and straw is in my opinion the best material for keeping you cool in the sun by letting your sweat easily evaporate.
The photo below is inside Muchelney Abbey where in 2002 I was playing a monk as part of a living history event.
Straw hats are unfortunately easily damaged so care must be taken in storage and transport, this one despite being 20 odd years old is still in good condition.
The only thing I added was tight string to keep it tied to head since even the lightest breeze can blow it off the head.
This hat was purchased at Tewkesbury Medieval Festival.