Naturally. No true cowgirl would be caught dead at a shindig without her hat... and no true cowgirl would let her hat get crushed.

Now for some unsolicited off-topic advice: Although I don't really understand why you find it embarassing that your mom wore one or two cowboy hats on the plane (she sounds like a very practical lady to me), if you want to do her and her hat(s) a favor and save yourself some embarassment you could buy her a hat can. She'll probably be more comfortable on the plane if she doesn't have to wear the hat constantly and the hat will be much safer in the overhead - or under the seat - if it's in it's own little carrier.