Mixture-of-Experts (MoE) inference with expert parallelism (EP) relies on All-to-All communication, making computation-communication overlap strategies critical for achieving high performance. However, no strategy is universally optimal. Across 103 production configurations, varying input lengths, and four hardware platforms, we observe that the fastest strategy shifts based on model shape, EP degree, input length, and hardware. Since no strategy dominates more than 80.6% of configurations, configuration-aware selection is necessary. In this work, we propose OverlapAdvisor, which models overlap stages using two granularities to capture computation-communication dependencies. OverlapAdvisor builds a unified performance model, accounting for deviations from peak compute, memory, and communication performance across workloads and hardware. This model enables launch-time, configuration-aware strategy selection within two serving engines. We conduct experiments on four MoE models (30B-235B) against three baselines, utilizing SGLang and vLLM. At the kernel level, OverlapAdvisor achieves 1.16x average and 1.59x maximum speedups, with a prediction MAPE of 3.3%-15.9%. In SGLang and vLLM, it improves end-to-end throughput up to 31.7%. Its granularity-guided optimizations further achieve median speedups of 1.10x for DeepEP-v1 and 1.22x for FlashMoE.

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