This paper identifies the cause of slow convergence for optically thick coarse mesh cells, when coarse mesh-based acceleration methods known in the literature are applied to the neutron transport criticality calculation. To overcome the limitation, this paper introduces two two-level iterative schemes to speed up coarse mesh-based acceleration, and they are applied to the partial current-based coarse mesh finite difference (p-CMFD) acceleration method. In the first scheme, a type of fine mesh finite difference (p-FMFD)- or intermediate mesh finite difference (p-IMFD)-based acceleration with a fixed fission source is augmented in a coarse mesh-based acceleration with power iteration. The second scheme applies global/local inner iterations in addition to the first scheme. Because p-CMFD is unconditionally stable and provides transport partial currents (instead of net current) on the interface between two coarse mesh cells, this enables the two schemes to speed up convergence even in optically thick coarse mesh cells. Numerical results on one-dimensional and two-dimensional test problems show that the two schemes (in particular, the scheme with global/local iterations) enhance the convergence speed of p-CMFD acceleration, especially for optically thick coarse mesh cells.