Compared with those of hydrodynamics, the existence of magnetic field leads to extra characteristic waves for magnetohydrodynamics (MHD), and further leads to an inconsistency of jump condition across the contact discontinuity. Usually, for the magnetic field variables in an HLLC Riemann solver, a single HLL intermediate state is used to replace two HLLC intermediate states to achieve conservation and computational stability, at the cost of insufficient simulation accuracy for tangential discontinuity. In this paper, a previouly developed HLLC solver is specially constructed to deal with MHD tangential discontinuities accurately and satisfy the so-called Toro condition. With numerical tests, such as the time-dependent simulation of one-dimensional shock tube, the tangential discontinuities, and the global MHD simulation of Earth's magnetosphere, we compare numerical results of the modified HLLC solver with those of the standard HLLC and HLLD solvers. It indicates that the modified HLLC solver has better capture accuracy for tangential discontinuities than the previously developed HLLC solver, and has the accuracy of the more time-consuming HLLD solver in some situations.