Pre-war Summer Rain Hat?

I have recently purchased a vintage Tilley-like Brooks cotton hat, which is, as suggested in other threads, ideal for trad styles on rainy days in summer. Then I have come to think what could have been an option in the days when everyone wore hats with suits, especially in the 20s, 30s and 40s. I am not sure if in those days there were "summer rain hats" in the first place, but boaters and wool fedoras are not effective against rain and humidity respectively. As my Google image search does not give me a clue, I would like to call for any thoughts from knowledgeable members here. What did they pick up to be worn with navy or gray suits on rainy summer days? I really appreciate your help.