When I saw her garb, the first thing that came to mind was she looked like a femfatal from a Spaghetti Western, so I looked up why she wore the hat.
The outfit is one she chose, and is an awful lot alike one worn by a character called Carmen Sandiego,
The hat, first one, was specially made for her, and -- some how -- horribly crushed when shipped to her, HMMM, so the builder had one day to make another.
This one was touched by no one but him until delivered to Melania.