The sense of touch can provide a robot with a wealth of information about the contact region when interacting with an unknown environment. Nevertheless, utilizing touch information to plan exploration paths and adjust robot posture to improve task efficiency remains challenging. This paper presents a novel approach for the online tactile surface exploration of unknown objects with a multidegree of freedom robotic hand. We propose an exploration strategy that actively maximizes the entropy of the acquired data while dynamically balancing the exploration's global knowledge and local complexity. We demonstrate that our method can efficiently control a multi-fingered robotic hand to explore objects of arbitrary shapes (e.g., with a handle, hole, or sharp edges). To facilitate efficient multi-contact exploration with a robotic hand, we offer an optimization-based planning algorithm that adapts the hand pose to the local surface geometry online and increases the kinematic configuration of each finger during exploration. Ultimately, we compared our approach to state of the art in a simulated environment. Experimental results indicate that our proposed methods can guide a multi-finger robotic hand to explore efficiently and smoothly, thereby reconstructing the unknown geometry of a variety of everyday objects, with significant improvements in data efficiency and finger compliance when compared to state-of-the-art approaches. (c) 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).