Previous design methodologies for spring-driven jumping robots focused on jump height optimization for specific tasks. In doing so, numerous designs have been proposed including using nonlinear spring-linkages to increase the elastic energy storage and jump height. However, these systems can never achieve their theoretical maximum jump height due to taking off before the spring energy is fully released, resulting in an incomplete transfer of stored elastic energy to gravitational potential energy. This paper presents low-order models aimed at characterising the energy conversion during the acceleration phase of jumping. It also proposes practical solutions for increasing the energy efficiency of jumping robots. A dynamic analysis is conducted on a multibody system comprised of rotational links, which is experimentally validated using a physical demonstrator. The analysis reveals that inefficient energy conversion is attributed to inertial effects caused by rotational and unsprung masses. Since these masses cannot be entirely eliminated from a physical linkage, a practical approach to improving energy efficiency involves structural redesign to reduce structural mass and moments of inertia while maintaining compliance with structural strength and stiffness requirements.