In [Z. Hu, R. Li, and Z. Qiao. Acceleration for microflow simulations of high-order moment models by using lower-order model correction. J. Comput. Phys., 327:225-244, 2016], it has been successfully demonstrated that using lower-order moment model correction is a promising idea to accelerate the steady-state computation of high-order moment models of the Boltzmann equation. To develop the existing solver, the following aspects are studied in this paper. First, the finite volume method with linear reconstruction is employed for high-resolution spatial discretization of the resulting equations. Second, by introducing an appropriate parameter τ in the correction step, it is found that the performance of the solver can be improved significantly, i.e., more levels would be involved in the solver, which further accelerates the convergence of the method. Third, Heun’s method is employed as the smoother in each level to enhance the robustness of the solver. Numerical experiments in microflows are carried out to demonstrate the efficiency and to investigate the behavior of the new solver. In addition, several order reduction strategies for the choice of the order sequence of the solver are tested, and the strategy ml−1 = ml/2 is found to be most efficient.