Real-time spoken dialogue systems face a fundamental tension between latency and response quality. End-to-end speech-to-speech (S2S) models respond immediately and naturally handle turn-taking, backchanneling, and interruption, but produce semantically weaker outputs. Cascaded pipelines (ASR -> LLM) deliver stronger responses at the cost of latency that grows with model size. We present RelayS2S, a hybrid architecture that runs two paths in parallel upon turn detection. The fast path - a duplex S2S model - speculatively drafts a short response prefix that is streamed immediately to TTS for low-latency response onset, while continuing to monitor live audio events. The slow path - a cascaded ASR -> LLM pipeline - generates a higher-quality continuation conditioned on the committed prefix, producing an uninterrupted utterance. A lightweight learned verifier gates the handoff, committing the prefix when appropriate or falling back gracefully to the cascaded pipeline. With GPT-4.1 as the back-end, RelayS2S substantially reduces response latency while preserving nearly all of the cascaded pipeline's textual quality. On synthetic voice dialogues, it achieves a P90 first-chunk latency of 81 ms, excluding TTS and network latency, compared with 1,006 ms for the cascaded baseline. On real voice dialogues, RelayS2S reduces average first-chunk latency by 479 ms while retaining 99% of the cascaded pipeline's textual quality. These benefits become larger as the slow-path model scales. Because the prefix handoff requires no architectural modification to either component, RelayS2S serves as a lightweight, drop-in addition to existing cascaded pipelines. Our code is publicly available at: https://github.com/mailong25/relays2s

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