Community detection is a classical problem in the field of graph mining. We are interested in local community detection where the objective is the recover the communities containing some given set of nodes, called the seed set. While existing approaches typically recover only one community around the seed set, most nodes belong to multiple communities in practice. In this paper, we introduce a new algorithm for detecting multiple local communities, possibly overlapping, by expanding the initial seed set. The new nodes are selected by some local clustering of the graph embedded in a vector space of low dimension. We validate our approach on real graphs, and show that it provides more information than existing algorithms to recover the complex graph structure that appears locally.