Recently, a three-dimensional method of characteristics (MOC) code called Advanced Reactor CHaracteristics tracER (ARCHER) has been developed by the Institute of Nuclear and New Energy Technology, Tsinghua University, to solve the neutron transport problem in high-temperature gas-cooled reactors (HTRs) with explicit pebble-bed geometry. Although the spatial domain decomposition using the message passing interface (MPI) and the ray parallel using OpenMP have been implemented in the previous version of ARCHER, in order to simulate practical HTR problems it is still necessary to reduce the great computational burden through efficient algorithms. Therefore, the linear source approximation (LSA) scheme, which allows coarser transport calculation grids while maintaining high accuracy, has been added in the latest version of ARCHER to relieve memory pressure together with the MPI-based spatial domain decomposition. Moreover, on-the-fly calculation of the relative position coordinates of the ray segment center can further reduce the memory for storing segment information under LSA. In addition, time-consuming MOC transport sweeps can be reduced greatly with coarse-mesh finite difference (CMFD) acceleration. Numerical results show that both LSA and CMFD acceleration contribute to simulate the practical HTR-10 problem successfully.