My understanding was that it depended upon the type of establishment, what you were trying to do there, and if there was a place to put your hat. Otherwise, the convention seemed to be that you generally could keep your hat on, unless it was in an elevator...for whatever reason, every man was expected to take off his hat in the elevators without exception. Weirdly, women were not ever really expected to remove their hats, and wearing hats indoors was considered dressing up for them. Women were allowed to wear fashion statement hats even in court...reminds me of how our pal Hulk Hogan had to get special permission from the judge in modern times to wear his signature bandanna in court.