Hierarchical Navigable Small World (HNSW) is widely adopted for approximate nearest neighbor search (ANNS) for its ability to deliver high recall with low latency on large-scale, high-dimensional embeddings. The exploration factor, commonly referred to as ef, is a key parameter in HNSW-based vector search that balances accuracy and efficiency. However, existing systems typically rely on manually and statically configured ef values that are uniformly applied across all queries. This results in a distribution-agnostic configuration that fails to account for the non-uniform and skewed nature of real-world embedding data and query workloads. As a consequence, HNSW-based systems suffer from two key practical issues: (i) the absence of recall guarantees, and (ii) inefficient ANNS performance due to over- or under-searching. In this paper, we propose Adaptive-ef (Ada-ef), a data-driven, update-friendly, query-adaptive approach that dynamically configures ef for each query at runtime to approximately meet a declarative target recall with minimal computation. The core of our approach is a theoretically grounded statistical model that captures the similarity distribution between each query and the database vectors. Based on this foundation, we design a query scoring mechanism that distinguishes between queries requiring only small ef and those that need larger ef to meet a target recall, and accordingly assigns an appropriate ef to each query. Experimental results on real-world embeddings produced by state-of-the-art Transformer models from OpenAI and Cohere show that, compared with state-of-the-art learning-based adaptive approaches, our method achieves the target recall while avoiding both over- and under-searching, reducing online query latency by up to 4x, offline computation time by 50x, and offline memory usage by 100x.