Speculative decoding accelerates Large Language Model inference via a draft-then-verify paradigm, yet the output projection layer becomes a bottleneck as vocabulary sizes scale. While existing static pruning methods effectively reduce this overhead, they suffer from precipitous drops in acceptance rate in specialized domains or topic-switching scenarios due to their inability to capture dynamic distribution shifts. To address this, we introduce EvoSpec, a framework that enables real-time evolution of the draft model through dynamic vocabulary and parameter adaptation. Unlike static or purely retrieval-based approaches, EvoSpec employs a context-aware mechanism that retrieves critical long-tail tokens via efficient semantic and statistical indexing. Furthermore, we propose a lightweight online alignment strategy utilizing curriculum learning to continually minimize the distributional gap between the draft and target models. Extensive evaluations across specialized domains (coding, law, and medicine) confirm that EvoSpec overcomes the limitations of static baselines. On EAGLE-3, it achieves a 1.13x speedup in these settings over the state-of-the-art static baseline FR-Spec, with 27\% lower memory overhead than standard online adaptation.