LLM agents are becoming a common interface for research, coding, and question answering, yet their Thought-Action-Observation loop is often serial: the model reasons, emits a tool call, then idles the GPU until the result returns. This wait consumes 16-37% of wall time in our workloads and 35-61% in prior reports. Speculative tool execution can hide this wait, but existing systems need auxiliary predictors, historical traces, or static workflow graphs, leaving a gap for training-free, day-one deployment. We observe that the model can be its own predictor: a probe forked at the start of generation predicts Qwen3-32B's upcoming tool name with 74.6-99.6% accuracy across five benchmarks. We present SPORK (Self-sPeculative fORKing), a training-free controller that dispatches the speculated tool call early, overlapping its execution with the remaining chain-of-thought decode. A cost model captures when speculation breaks even, and each component improves one of its terms: a prefix-cache fork cuts probe cost, a confidence gate filters mispredictions, and partial-token accept turns rejected probes into speculative-decoding drafts. On acceptance, the tool result is ready when reasoning ends; on rejection, SPORK falls back to serial execution with no correctness penalty. On real-tool benchmarks, SPORK cuts Qwen3-32B's GAIA P95 by 18% (131.9 to 108.1 s); the mechanism holds across model sizes from 4B to 32B and across dense and mixture-of-experts models, with task accuracy within 1 pp of baseline or better wherever measured. SPORK deploys as a thin controller over standard completion APIs (no retraining, no auxiliary models, no offline traces) and is orthogonal to token-level speculative decoding. SPORK is open source at https://github.com/baihuajun24/spork.