We consider manipulation problems in constrained and cluttered settings, which require several regrasps at unknown locations. We propose to inform an optimization-based task and motion planning (TAMP) solver with possible regrasp areas and grasp sequences to speed up the search. Our main idea is to use a state space abstraction, a regrasp map, capturing the combinations of available grasps in different parts of the configuration space, and allowing us to provide the solver with guesses for the mode switches and additional constraints for the object placements. By interleaving the creation of regrasp maps, their adaptation based on failed refinements, and solving TAMP (sub)problems, we are able to provide a robust search method for challenging regrasp manipulation problems.