This paper presents Rummaging Using Mutual Information (RUMI), a method for online generation of robot action sequences to gather information about the pose of a known movable object in visually-occluded environments. Focusing on contact-rich rummaging, our approach leverages mutual information between the object pose distribution and robot trajectory for action planning. From an observed partial point cloud, RUMI deduces the compatible object pose distribution and approximates the mutual information of it with workspace occupancy in real time. Based on this, we develop an information gain cost function and a reachability cost function to keep the object within the robot's reach. These are integrated into a model predictive control (MPC) framework with a stochastic dynamics model, updating the pose distribution in a closed loop. Key contributions include a new belief framework for object pose estimation, an efficient information gain computation strategy, and a robust MPC-based control scheme. RUMI demonstrates superior performance in both simulated and real tasks compared to baseline methods.