“Former hat” is not idiomatic but “hat she used to wear” is, and that is what “ancien chapeau” means. It remains to find a word or at least a shorter combination of English words to express that, if possible. I know of none that should be unambiguous. “Previous hat” is not better than “former hat”.

You can say “old” in English though, as in “my old school”; it means “former” as you’ll probably recall, but if “former” is not idiomatic for “hat”, I think that “old” is (OALD, number 5).

• She is selling an old hat of hers.