Haptic exploration in robotics is prone to sensing ambiguities. Actively selecting actions during exploration can provide crucial information to mitigate these ambiguities and improve object recognition. This study presents a dual-stage active haptic exploration technique that enables a robot to adapt its actions to optimise information acquisition for object recognition. In the initial stage of rough perception, the algorithm employs actions that maximise mutual information to swiftly identify the likely categories of an object. Subsequently, during the fine perception stage, it selects actions that maximise the Kullback-Leibler (KL) divergence between the most likely pair of ambiguous objects, thus facilitating their differentiation. To evaluate the performance of our algorithm, a robot with a sensorised finger collected tactile information from the interaction with ten objects using the primary actions of pressing, sliding, and tapping. In comparison with existing active exploration strategies that optimise a single information metric, our algorithm achieves superior recognition rates while requiring fewer exploration actions. By conducting only necessary comparisons between similar objects, it also reduces the computational cost. These results suggest that the proposed algorithm effectively diminishes ambiguities by adapting actions and enhancing the recognition outcomes in haptic exploration.