When I lived in NYC in 1979 and started my first job, I was kind of surprised to see so many guys wearing hats, but only men who went into the building where I worked.
Took me a few days to realize The Cloth Hat, Cap and Millinery Workers International Union (CHCMW) was on the floor above where I worked.
From that experience, I knew exactly what hat you were referring to in the OP.
Don’t know if that union still exists, or if they still have an entire floor in that NYC building, but I certainly saw lots of styles of hats for men on guys going in and out of the building, but you could tell it was sadly a dying trade in the city.