To address the narrow convergence basin and high sensitivity to initial guesses of the incremental harmonic balance method in the analysis of strongly nonlinear systems, this paper develops an enhanced approach that combines a trust region strategy with asymptotic expansion. In each incremental step, an artificial asymptotic parameter is introduced, and the nonlinear incremental equations are expanded into a series of linear recursive subproblems based on the Hessian matrix, enabling order-by-order decoupling of the response components. Furthermore, a trust region algorithm is employed as the inner solver to adaptively control the iteration step size, ensuring robustness and global convergence in solving each subproblem at each order. Numerical examples demonstrate that the proposed method preserves the accuracy of the traditional approach, while significantly widening the convergence basin, reducing dependence on initial guesses, and tracing the frequency response curve with fewer iterations and less computational time. The method, therefore, provides a semi-analytical framework with robust convergence and high computational efficiency for periodic response analysis of strongly nonlinear systems.