It is well known that the FGMRES algorithm can be used as an alternative to iterative refinement and, in some instances, is successful in computing a backward stable solution when iterative refinement fails to converge. In this study, we analyse how variants of the Chebyshev algorithm can also be used to accelerate iterative refinement without loss of numerical stability and at a computational cost at each iteration that is less than that of FGMRES and only marginally greater than that of iterative refinement. A component-wise error analysis of the procedure is presented and numerical tests on selected sparse test problems are used to corroborate the theory.