This paper investigates the use of local search strategies to improve the performance of a meta-hyper-heuristic algorithm, a hyper-heuristic which employs one or more meta-heuristics as low-level heuristics. Alternative mechanisms for selecting the solutions to be refined further by means of local search, as well as the intensity of subsequent refinement in terms of number of allowable function evaluations, are investigated. Furthermore, defining a local search as one of the low-level heuristics versus applying the algorithm directly to the solution space is also investigated. Performance is evaluated on a diverse set of floating-point benchmark problems. The addition of local search was found to improve algorithm results significantly. Random selection of solutions for further refinement was identified as the best selection strategy and a higher intensity of refinement was identified as most desirable. Better results were obtained by applying the local search algorithm directly to the search space instead of defining it as a low-level heuristic.