To address the nonlinear characteristics, strong coupling, and numerical instability near singular configurations in inverse kinematics solving of six-degree-of-freedom manipulators, a hybrid solution method combining Adaptive Mutation Particle Swarm Optimization and the Levenberg–Marquardt algorithm, namely AM-PSO-LM, is proposed. A unified normalized residual model is constructed using unit-quaternion-based orientation error and position error, thereby ensuring the consistency of the objective function definition between the global search stage and the local refinement stage, and avoiding deviation of the optimization direction during stage transition. A probing-based switching strategy is introduced to realize a smooth transition from particle swarm search to LM local refinement, and a hierarchical rescue mechanism after local refinement failure is further incorporated to improve the solution stability in the neighborhood of singular configurations. Taking the PUMA560 six-degree-of-freedom manipulator as the research object, 500 independent simulation tests are carried out, and comparative analyses with several algorithms are performed under both regular and singular conditions. The results show that the success rates of AM-PSO-LM reach 82.6% and 91.6% under regular and singular conditions, respectively. The median position error and orientation error are lower than 1.00 × 10?? mm and 1.00 × 10??°, respectively, and the average computation time for a single solution is 0.040 s. The proposed method achieves coordinated improvement in solution accuracy, computational efficiency, and robustness near singular configurations, providing an effective hybrid optimization approach for high-precision inverse kinematics solving of six-degree-of-freedom manipulators.