Touching your Brim in Greeting.....

I've never read about it being a thing ladies did. For men, it relates to the military salute (I have a half-remembered memory that the salute developed aping the tipping of the hat rather than the other way around, but I' sure someone better versed can correct me here). I will occasionally do it as a greeting myself, though a considered bow of the head as a sideways angle comes more naturally to me.

 

I always forget and just give the dorky nod I always have.

We have a whole thread on hat etiquette around here somewhere. That might be it over there under that table...

 

It wasn't customary for women to tip hats -- they were usually pinned in place anyway -- so there was no need to touch the brim in any kind of salute, assuming the hat being worn even had a brim. Women also weren't expected to remove their hats in church or when saluting the flag or even when eating in a restaurant. You better take it off at the movies though, or some jerk behind you will start kicking your seat.
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Ditto at baseball games. In 1942's "Woman of the Year," an oblivious large-hat-wearing Katherine Hepburn is called out several times by the men sitting behind her.

 

Boy, that Frankie Germano gets around.