Graph indexes are widely used for high-recall approximate nearest neighbor search (ANNS), but many real-time applications require streaming ANNS. In these real-time applications, continuously arriving embeddings must search the existing graph for candidate neighbors before updating graph edges, which makes repeated index construction a bottleneck for streaming ingestion workloads. We propose Slipstream, a new method that significantly reduces the computational cost of frequent insertions in graph indexes for ANNS. The core idea of Slipstream is exploiting the continuity in vector streams: the newly arrived point starts from promising candidates found during the previous insertion rather than searching from the entry point. More technically, Slipstream evaluates distinct subsets of starting candidates followed by an adaptive controller that narrows or widens the range according to the stream's stability. We further show that Slipstream is beyond heuristic: We derive an abstract model to characterize Slipstream's performance and analyze its theoretical bounds. We have implemented Slipstream in two popular open-source libraries (Faiss, HNSWLib) and compared it with four baseline methods on five streaming vector datasets. Experimental results show that Slipstream achieves up to 30.8$\times$ higher end-to-end throughput than baselines while maintaining at least 0.95 recall@10.