Ranchgal - Thanks for the comments about the hats. I think that is what they must have been referring to. (Are you saying that women were held to the same "remove it out of respect" standards that men were? I find that interesting.)
No they never really got as picky with the girls-but they didn't consider a cowboy hat a bonnet either--but when in people buildings I just did anyway, especially at cattle conferences and meetings, when you would probably be around both types,LOL and usually tried to leave mine at home. Just made everything easier if I wasn't worried about what to do with it.
NO when women met people they were allowed to keep their hats on, anytime except church and the national anthem.