Recently, a series of discontinuity-capturing schemes with the boundary variation diminishing (BVD) algorithm are proposed. The BVD algorithm requires that the jump of reconstructed values at a cell boundary should be minimized so as to reduce the numerical dissipation in Riemann solvers. This paper explores the possible reason for the spurious overshoots generated by multi-stage BVD algorithms in solving several gas dynamics problems. Through analyzing the dissipation characteristics of the typical Riemann solvers combined with boundary variation, we propose a modified BVD algorithm to suppress the spurious overshoots occured in BVD schemes. Numerical results demonstrate that the novel algorithm can suppress the spurious overshoots while preserving the advantages of BVD schemes for contact discontinuities and smooth extrema.