The p-next center problem (pNCP) is an extension of the classical p-center problem. It consists of locating p centers from a set of candidate centers and allocating both a reference and a backup center to each client, to minimize the maximum cost, which is the length of the path from a client to its reference center and then to its backup center. Among them, the reference center is the closest center to a client and serves it under normal circumstances, while the backup center is the closest center to the reference center and serves the client when the reference center is out of service. In this paper, we propose a weighting-based tabu search algorithm called WTS for solving pNCP. WTS optimizes the pNCP by solving its decision subproblems with given assignment costs with an efficient swap-based neighborhood structure and a hierarchical penalty strategy for neighborhood evaluation. Extensive experimental studies on 413 benchmark instances demonstrate that WTS outperforms the state-of-the-art methods in the literature. Specifically, WTS improves 12 previous best known results and matches the optimal results for all remaining 401 ones in a much shorter time than other algorithms. More importantly, WTS reaches the lower bounds for 10 instances for the first time.