We address the design of optimal strategies for a pursuer trying to catch a moving evader. When the pursuer has available two teams of agents with different capabilities-one that can "search for the evader" and the other one that can "catch the evader"-, the game can be naturally formulated as an optimal control problem on a hybrid system. We show that solving the hybrid pursuit game is equivalent to finding a Stackelberg equilibrium solution for a partial information Markov game, which can be solved using dynamic programming. Since for most realistic situations this approach is computationally very difficult, we propose a two-level suboptimal solution that uses a greedy control for coordinating the agents within each team, and a threshold-based logic for orchestrating the switching between teams. Simulations are included to show the feasibility of the approach.