The set covering problem (SCP) is a fundamental NP-hard problem in computer science and has a broad range of important real-world applications. In practice, SCP instances transformed from real-world applications would be of large scale, so it is of significant importance to design effective heuristic algorithms, especially local search ones. However, there exist only few research works on developing local search algorithms for solving SCP. In this article, we propose a new local search algorithm for solving SCP, dubbed NuSC. In particular, NuSC introduces a new combined scoring function for subset selection, which combines different subset properties in an effective way and helps NuSC find more optimized solutions. Besides, NuSC incorporates a dynamic weighting scheme for elements, a tabu search strategy, and a novelty selection mechanism to further enhance its practical performance. In order to study the effectiveness and robustness of our proposed NuSC algorithm, we conduct extensive experiments to compare NuSC against many state-of-the-art competitors on various types of SCP instances. Our experimental results demonstrate that NuSC significantly outperforms its competitors on the majority of instances, indicating the superiority of NuSC. Also, our empirical evaluations confirm the effectiveness of each algorithmic technique underlying NuSC.