Length-aware scheduling reduces request latency in LLM serving by prioritizing shorter requests. However, despite their large model sizes, length predictors can suffer degraded ranking performance under domain and task shifts, and adapting them can require costly retraining. Their prediction overhead can further offset the benefits of scheduling. To improve the generalization and efficiency of length-aware scheduling, we propose \textbf{LEAP}, a length prediction system with fast online adaptation and asynchronous CPU execution. LEAP reuses the serving model's prefill hidden states, which contain information about the output length, and extracts this signal through a linear readout. To adapt to substantial domain and task shifts, it updates the readout at test time through recursive least squares, using completed requests as supervision without gradient computation or historical-data replay. Prediction and updates run asynchronously on CPU, while prediction and initial batch ordering overlap the remaining GPU prefill. Across the evaluated cross-domain transfers, LEAP retains strong ranking performance, and online adaptation improves Spearman correlation by 64.7\% under a larger domain shift. Serving experiments show that LEAP reduces average per-token latency by 15.9--40.8\% compared with the evaluated length predictors, while achieving 1.9\% higher throughput with $30{,}569\times$ fewer predictor parameters than the strongest evaluated baseline.

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