Legged robots offer several advantages when navigating unstructured environments, but they often fall short of the efficiency achieved by wheeled robots. One promising strategy to improve their energy economy is to leverage their natural (unactuated) dynamics using elastic elements. This work explores that concept by designing energy-optimal control inputs through a unified, multi-stage framework. It starts with a novel energy injection technique to identify passive motion patterns by harnessing the system's natural dynamics. This enables the discovery of passive solutions even in systems with energy dissipation caused by factors such as friction or plastic collisions. Building on these passive solutions, we then employ a continuation approach to derive energy-optimal control inputs for the fully actuated, dissipative robotic system. The method is tested on simulated models to demonstrate its applicability in both single- and multi-legged robotic systems. This analysis provides valuable insights into the design and operation of elastic legged robots, offering pathways to improve their efficiency and adaptability by exploiting the natural system dynamics.