You should be able to uncover something about Caroline's life in New Orleans by locating the will/succession for Jacques Puech. It would list what he owned, how much it was worth, etc. If he left a will, he probably mentioned when he left Saint-Domingue. The estate papers for Orleans Parish are online at pilot.familysearch.org. If you already know when he came over, have you compared it to when Caroline was born? Was she born in Saint-Domingue? Good luck!