There has been a growing academic interest in the recognition of nested named entities in many domains. We tackle the task with a novel local hypergraph-based method: We first propose start token candidates and generate corresponding queries with their surrounding context, then use a query-based sequence labeling module to form a local hypergraph for each candidate. An end token estimator is used to correct the hypergraphs and get the final predictions. Compared to span-based approaches, our method is free of the high computation cost of span sampling and the risk of losing long entities. Sequential prediction makes it easier to leverage information in word order inside nested structures, and richer representations are built with a local hypergraph. Experiments show that our proposed method outperforms all the previous hypergraph-based and sequence labeling approaches with large margins on all four nested datasets. It achieves a new state-of-the-art F1 score on the ACE 2004 dataset and competitive F1 scores with previous state-of-the-art methods on three other nested NER datasets: ACE 2005, GENIA, and KBP 2017.