1) Winged cap of Hermes 2) Winged hat of Hermes 
A petasos or petasus (πέτασος) is a sun hat of Thessalian origin worn by the ancient Greeks, often in combination with the chlamys cape. It was usually made of wool felt, leather or straw, with a broad, floppy brim. It was worn primarily by farmers and travellers, and was considered characteristic of rural people. As a winged hat, it became...
hat.
wide-brimmed hat with a conical crown worn in ancient Greece. The petasos worn by men had a rather low crown, while that worn by women had a tall one.
a Thessalian broad-brimmed hat.