This paper introduces a novel model-free and partially model-free algorithm for inverse optimal control, also known as inverse reinforcement learning, aimed at estimating the cost function of continuous-time nonlinear deterministic systems. Using the input-state trajectories of an expert agent, the proposed algorithm separately utilizes control policy information and the Hamilton-Jacobi-Bellman equation to estimate different sets of cost function parameters. This design allows the framework to achieve broader applicability while maintaining a model-free formulation. Moreover, the model-free version reduces computational complexity compared to existing methods, as it requires solving a forward optimal control problem only once during initialization. In contrast, in the partially model-free version, this step can be bypassed entirely for systems with known input dynamics. Simulation results demonstrate the effectiveness and efficiency of the proposed approach, highlighting its potential for real-world deployment in autonomous systems and robotics.