  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 so that the degrees of freedom in
spatial space could be reduced remarkably without loss of accuracy. Second, by
introducing an appropriate parameter $\tau$ 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
$m_{l-1} = \lceil m_{l} / 2 \rceil$ is found to be most efficient.
