Hi,
Hard to pin it down, as again it is a classic style that was certainly around in the 1930's as Grandmas Closet has stated. The type of glazed straw and the label give me a more 40's vibe....I think just around the WWII era, circa mid to late 1940s, leaning to post WWII. I think but am not 100% sure....and yes it goes to the back of the head.
The green hat looks a bit older, 1938-1940 with the puckered strap and a more natural type of straw. It is certainly not uncommon for them to have not changed the label during and after the war, they just used what they had in stock.
Dating hats is tricky.....sigh....I still struggle at it.
Gage was one of the major hat makers since the late 19th century. They sold both wholesale and retail, and had in house designers such as the woman who designed your hats. A Gage hat was a sign of a good hat for many years. I think they closed up in the late 1960s, but I cannot recall exactly now.
B