Entity set expansion, the task of expanding a small set of similar entities into a much larger set, is a vital step for downstream tasks such as named entity recognition, knowledge base construction and information retrieval. Existing entity set expansion methods were developed by mainly considering entities at coarse-grained levels, which encounter difficulties for entity set expansion at fine-grained levels, due to the subtlety on fine-grained type inference and semantic drifting. In this study, we propose an automated (i.e. without human annotation), fine-grained set expansion framework, FGExpan, which utilizes a taxonomy structure and a pre-trained language model to achieve high performance. To facilitate our testing, a new fine-grained set expansion dataset is also constructed. Experiments on this dataset and those used in previous studies show that FGExpan achieves significantly better performance (MAP up by 0.176) on fine-grained types and also the state-of-the-art expansion quality on coarse-grained entity sets.