Lunar tubes, natural underground structures on the Moon formed by ancient volcanic activity, offer natural protection from extreme temperatures, radiation, and micro-meteorite impacts, making them prime candidates for future lunar bases. However, the exploration of lunar tubes requires a high degree of mobility. Given the Moon's gravity, which is approximately six times weaker than Earth's, efficient navigation across rugged terrains within these lava tubes is achievable through jumping. In this work, we present the design of subsystems for a miniature hexapod rover weighing 1 kg, which can walk, jump, and stow. The walking system consists of two subsystems: one for in-plane walking, employing four single-degree-of-freedom (DoF) legs utilizing the KLANN walking mechanism, and another for directional adjustments before jumping. The latter employs a novel three-DoF mechanism with the cable-pulley system to optimize space utilisation. The design of these legs prioritizes functionality, minimizing the number of actuators and mass. Furthermore, we introduce a jumping mechanism featuring a spring-based energy storage system and a sector-gear-based release mechanism. This mechanism is engineered to enable jumps with a maximum horizontal range of 1 meter on the Moon. It has been 3D printed and independently demonstrated. Our work encompasses a comprehensive system-level design study, with a focus on sub-system-level functions for various mechanisms. The innovative mechanisms designed and demonstrated in this work hold potential for broader aerospace applications, including payload separation systems, pushing mechanisms, and door opening and closing mechanisms, among others.