We consider a set of challenging sequential manipulation puzzles, where an agent has to interact with multiple movable objects and navigate narrow passages. Such settings are notoriously difficult for Task-and-Motion Planners, as they require interdependent regrasps and solving hard motion planning problems. In this paper, we propose to search over sequences of easier pick-and-place subproblems, which can lead to the solution of the manipulation puzzle. Our method combines a heuristic-driven forward search of subproblems with an optimization-based Task-and-Motion Planning solver. To guide the search, we introduce heuristics to generate and prioritize useful subgoals. We evaluate our approach on various manually designed and automatically generated scenes, demonstrating the benefits of auxiliary subproblems in sequential manipulation planning.