To reduce the numerical dissipation in turbulence modeling while maintaining the numerical stability around flow discontinuities in supersonic flowfield, a low-dissipative compressible solver is developed for large-eddy simulation within the OpenFOAM framework. To achieve the aforementioned goals, the low-dissipative solver adopts the hybrid scheme approach through combining the dissipative Kurganov-Tadmor scheme with the nondissipative central scheme via a shock sensor. In the construction of the central scheme, a robust skew-symmetric form of the convective term is adopted to preserve the local kinetic energy without adding an explicit dissipative term. Another feature of the low-dissipative solver is the implementation of an optimal explicit strong stability-preserving linear third-order total variation diminishing Runge-Kutta method for the temporal discretization. Numerical tests for a series of canonical flow problems are carried out to validate the solver's good performance in the flowfield either with strong discontinuities or with continuous spectrum characteristics. Large-eddy simulation of a scramjet combustor with supersonic airstream passing over the flame holder is conducted to validate the low-dissipative solver's reliability in a realistic flow with the complex interaction of shock discontinuities and turbulence.