What hat? The petasos? I don't know exactly, no findings so far of which I know. I'm momentarily not aware of literary evidence, but there may be some. It is highly likely that it was made from different materials, felt, leather and even straw in some cases. It is a mystery how the strange shape of some of the hats was achieved.
The pilos cap was presumably made most often from felt (pilos means "felt", "made of felt" in ancient Greek), but perhaps were also made sometimes from leather when you look at some ancient paintings. My petasos hat and my pilos cap are made from felt, the petasos is treated with glue to achieve the typical shape. To be honest, I don't like the result so much.

@ JRMC: thank you for your answer. I will put my trust in the EB historians of course, they cannot overlook the archaeological and pictoral evidence.