Catherine’s headgear is a cornette, which is a type of wimple. Believe it or not, her cornette was the ordinary headgear for a middle-class lady in France in the early 1800s. Her order, the Daughters of Charity, worked taking care of the poor and sick, and they tried to dress just like normal middle-class women in order to blend in with the local people. So instead of wearing a veil, they wore the cornette hat like lay women.
Obviously, over time the styles changed while the sisters’ habit did not, so the cornette became a distinctive feature of this order. They stopped wearing the cornette in the early 1960s.