In this paper we present two new approaches for finding good starting solutions to the planar p-median problem. Both methods rely on a discrete approximation of the continuous model that restricts the facility locations to the given set of demand points. The first method adapts the first phase of a greedy random construction algorithm proposed for the minimum sum of squares clustering problem. The second one implements a simple descent procedure based on vertex exchange. The resulting solution is then used as a starting point in a local search heuristic that iterates between the well-known Cooper’s alternating locate-allocate method and a transfer follow-up step with a new and more effective selection rule. Extensive computational experiments show that (1) using good starting solutions can significantly improve the performance of local search, and (2) using a hybrid algorithm that combines good starting solutions with a "deep" local search can be an effective strategy for solving a diversity of planar p-median problems.