Zero-shot cross-lingual transfer by fine-tuning multilingual pretrained models shows promise for low-resource languages, but often suffers from misalignment of internal representations between languages. We hypothesize that even when the model cannot generalize across languages effectively in fine-tuning, it still captures cross-lingual correspondence useful for cross-lingual transfer. We explore this hypothesis with Self-Translate-Train, a method that lets large language models (LLMs) to translate training data into the target language and fine-tunes the model on its own generated data. By demonstrating that Self-Translate-Train outperforms zero-shot transfer, we encourage further exploration of better methods to elicit cross-lingual capabilities of LLMs.