"...but it is also a World War I soldier's helmet with wings added."

Or, it could also be a pie-pan with wings added! Or a hubcap with wings added! MDR hit the nail on the head: the GA Flash's helmet is clearly an homage to the Greek/ Roman messenger-god who was often depicted with a winged helmet and wings on his ankles/ shoes. Thus, the GA Flash has a winged helmet and...wings on his shoes. Marvel's GA speedster the Whizzer also has wings on his helmet, but no wings on his ankles, while the Sub-Mariner does have wings on his ankles. Figure that one out.

The Flash's "hat" is, as has been said, based on the one worn by Mercury but it is also a World War I soldier's helmet with wings added.

The JSA is my favourite superhero group. The characters are more down to earth in their origin stories than the modern heroes, but at the same time more interesting. Dr Midnite was a physician blinded in an explosion who discovers that he can see but only in the dark. Hourman is a scientist who invents a pill that gives him super-strength, but only for an hour, and so on.

That "silly hat" is a winged helmet similar in style to the winged cap worn by Hermes. The same for the winged shoes. Looking forward to the JSA appearances on both shows.