Learned dense representations are a popular family of techniques for encoding queries and documents using high-dimensional embeddings, which enable retrieval by performing approximate k nearest-neighbors search (A-kNN). A popular technique for making A-kNN search efficient is based on a two-level index, where the embeddings of documents are clustered offline and, at query processing, a fixed number N of clusters closest to the query is visited exhaustively to compute the result set. In this paper, we build upon state-of-the-art for early exit A-kNN and propose an unsupervised method based on the notion of patience, which can reach competitive effectiveness with large efficiency gains. Moreover, we discuss a cascade approach where we first identify queries that find their nearest neighbor within the closest t << N clusters, and then we decide how many more to visit based on our patience approach or other state-of-the-art strategies. Reproducible experiments employing state-of-the-art dense retrieval models and publicly available resources show that our techniques improve the A-kNN efficiency with up to 5x speedups while achieving negligible effectiveness losses. All the code used is available at https://github.com/francescobusolin/faiss_pEE