Current exact single-linkage clustering algorithms have asymptotically quadratic complexity. We present algorithms for approximate single-linkage clustering with empirically near-linear scalability. We explore both graph index-based incremental nearest neighbor search and an iterative exploration scheme on the graph index approximating the MST of the reachability graph similar to Kruskal. As graph index, we use both the bottom layer and a combination of all layers of an HNSW as a stand-in for connected search graphs. We provide experiments comparing the clusterings to baselines such as exact single linkage implementation and an algorithm using metric tree-based searchers. We explore the impact of the HNSW hyperparameters on the performance in terms of running time and clustering quality and evaluate the empirical asymptotic complexity.