"un chapeau blanc" means "a white hat" in French.
Porter un chapeau = to wear a hat
chapeau
Un chapeau (masc.) is 'a hat' in English.
un chapeau
Un chapeau de paille is a straw hat in French.
Le vert chapeau
If you mean by French hat the kind artists typically wear, you are referring to a beret. A beret is a soft, round, flat-crowned hat, usually of woven, hand-knitted wool, crocheted cotton, wool felt, or acrylic fabric.
"un cheveu" (singular, plural: des cheveux) is a masculine noun in French.
un chat dans un chapeau (I'm sorry, it doesn't rhyme in French)
a fedora is called 'un chapeau mou' or 'un feutre mou' in French.
Je porte un chapeau
il porte un chapeau