Finding the densest subgraph (DS) from a graph is a fundamental problem in graph databases. It has been extensively studied in the literature and has found many real applications in a wide range of fields, such as biology, finance, and social networks. This paper studies how to efficiently discover the locally $h$-clique densest subgraph (L$h$CDS), which is a recently-proposed variant of DS. An L$h$CDS is a subgraph which is the densest among the ``local neighbors''. Given a graph $G$, a number of L$h$CDSes can be returned, which reflect different dense regions of $G$ and thus give more information than DS. Existing L$h$CDS solutions suffer from low efficiency due to extensively redundant computations. To improve efficiency, in this paper, we propose a divide-and-conquer-based algorithm, which not only reduces the search space but also has an improved time complexity. Extensive experiments on 15 real large graph datasets show that our proposed algorithm is up to two orders of magnitude faster than the state-of-the-art.