The goal of our research was to enhance local search heuristics used to construct Latin Hypercube Designs. First, we introduce the 1D-move perturbation to improve the space exploration performed by these algorithms. Second, we propose a new evaluation function ψp,σ specifically targeting the Maximin criterion. Exhaustive series of experiments with Simulated Annealing, which we used as a typically well-behaving local search heuristics, confirm that our goal was reached as the result we obtained surpasses the best scores reported in the literature. Furthermore, the ψp,σ function seems very promising for a wide spectrum of optimization problems through the Max-imin criterion.