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 scheme minimizes the residual norm of the multigrid iterate in a subspace and frequently achieve even faster convergence. The IMRS acceleration schemes reduce the cost of the standard MRS acceleration by about 40% for two dimensional problems. Numerical experiments are employed to demonstrate the efficiency of the IMRS acceleration schemes.