We introduce some inexact versions of the minimal residual smoothing (IMRS) technique to accelerate the standard multigrid convergence. These are modified versions of the minimal residual smoothing (MRS) technique introduced and analyzed in earlier papers. The IMRS acceleration schemes reduce the cost of the standard MRS acceleration by about 40% for two dimensional problems and frequently achieve even faster convergence. Numerical experiments are employed to demonstrate the efficiency of the IMRS acceleration schemes. Some of the numerical results show that IMRS can reduce the numbers of iterations by 88% with respect to the standard multigrid method.