Hi,
I realize that I never answered one of your initial questions. To answer: Grosgrain ribbons were standardized for use inside of ladies hats right around 1930. No hard and fast rule on this, but you don't see hats with grosgrain ribbons around the inside circumference until 1930. By the early 1930's lining had mostly disappeared in mass manufactured hats, although small milliners still used them for custom hats. Haute Couture, of course darling, still used full linings in hats for decades after that. Ahhhhh...the French!
In the 1930's, you will find hats with partial linings, full linings and grosgrain ribbons of various widths. But by 1933 or so, it was standard for just a ribbon inside.
I hope that helped!
By the way, I think the bow goes in front, at least from what I can determine from your interior photo. I am also looking at the type of straw, which looks to be a glazed natural straw that was popular from about the late 1930s to the 1950's. Of course, a milliner would use whatever she had, regardless of when the material was made or first popular. The label looks post WWII, and the fabric used for the band and bow looks post WWII also. With the bow in front, it gives it an even more early 1950s look. The label should go directly in back when wearing it.
I do think anyone could get away with wearing this with a 1930's or 1940's outfit, regardless of when it was made.