A new local search heuristic, called J-MEANS, is proposed for solving the minimum sum-of-squares clustering problem. The neighborhood of the current solution is defined by all possible centroid-to-entity relocations followed by corresponding changes of assignments. Moves are made in such neighborhoods until a local optimum is reached. The new heuristic is compared with two other well-known local search heuristics, K-MEANS and H-MEANS as well as with H-MEANS+, an improved version of the latter in which degeneracy is removed. Moreover, another heuristic, which fits into the Variable Neighborhood Search metaheuristic framework and uses J-MEANS in its local search step, is proposed too. Results on standard test problems from the literature are reported. It appears that J-MEANS outperforms the other local search methods, quite substantially when many entities and clusters are considered.