Non-monotone object rearrangement planning in confined spaces such as cabinets and shelves is a widely occurring but challenging problem in robotics. Both the robot motion and the available regions for object relocation are highly constrained because of the limited space. This work proposes a Multi-Stage Monte Carlo Tree Search (MS-MCTS) method to solve non-monotone object rearrangement planning problems in confined spaces. Our approach decouples the complex problem into simpler subproblems using an object stage topology. A subgoal-focused tree expansion algorithm that jointly considers the high-level planning and the low-level robot motion is designed to reduce the search space and better guide the search process. By fitting the task into the MCTS paradigm, our method produces optimistic solutions by balancing exploration and exploitation. The experiments demonstrate that our method outperforms the existing methods regarding the planning time, the number of steps, and the total move distance. Moreover, we deploy our MS-MCTS to a real-world robot system and verify its performance in different confined environments.