Robust supervised fine-tuned small Language Models (sLMs) often show high reliability but tend to undercorrect. They achieve high precision at the cost of low recall. Conversely, Large Language Models (LLMs) often show the opposite tendency, making excessive overcorrection, leading to low precision. To effectively harness the strengths of LLMs to address the recall challenges in sLMs, we propose Post-Correction via Overcorrection (PoCO), a novel approach that strategically balances recall and precision. PoCO first intentionally triggers overcorrection via LLM to maximize recall by allowing comprehensive revisions, then applies a targeted post-correction step via fine-tuning smaller models to identify and refine erroneous outputs. We aim to harmonize both aspects by leveraging the generative power of LLMs while preserving the reliability of smaller supervised models. Our extensive experiments demonstrate that PoCO effectively balances GEC performance by increasing recall with competitive precision, ultimately improving the overall quality of grammatical error correction.