Where: US large city
(I've tried several historic clothing sites but none are specific enough, and the various costuming lists have a long cycle time.)
My MC, a fashion-aware woman in her mid-twenties with limited middle class means, needs a stylish (or at least not too out-of-style) hat that has (or can have) a veil. Small is good; I'd use pillbox, but what I could find suggests though they existed since the 1930s, they weren't popular until much later in the 1950s.
A belated thanks to everyone who contributed. I don't have _the_ answer, but it's a nice place to start from.