Large Language Models (LLMs) have achieved impressive progress across a wide range of tasks, yet their heavy reliance on English-centric training data leads to significant performance degradation in non-English languages. While existing multilingual prompting methods emphasize reformulating queries into English or enhancing reasoning capabilities, they often fail to incorporate the language- and culture-specific grounding that is essential for some queries. To address this limitation, we propose EMCEE (Extracting synthetic Multilingual Context and merging), a simple yet effective framework that enhances the multilingual capabilities of LLMs by explicitly extracting and utilizing query-relevant knowledge from the LLM itself. In particular, EMCEE first extracts synthetic context to uncover latent, language-specific knowledge encoded within the LLM, and then dynamically merges this contextual insight with reasoning-oriented outputs through a judgment-based selection mechanism. Extensive experiments on four multilingual benchmarks covering diverse languages and tasks demonstrate that EMCEE consistently outperforms prior approaches, achieving an average relative improvement of 16.4% overall and 31.7% in low-resource languages.