The performance of Large Language Models (LLMs) is highly sensitive to the prompts they are given. Drawing inspiration from the field of prompt optimization, this study investigates the potential for enhancing Automated Essay Scoring (AES) by refining the scoring rubrics used by LLMs. Specifically, our approach prompts models to iteratively refine rubrics by reflecting on models' own scoring rationales and observed discrepancies with human scores on sample essays. Experiments on the TOEFL11 and ASAP datasets using GPT-4.1, Gemini-2.5-Pro, and Qwen-3-Next-80B-A3B-Instruct show Quadratic Weighted Kappa (QWK) improvements of up to 0.19 and 0.47, respectively. Notably, even with a simple initial rubric, our approach achieves comparable or better QWK than using detailed human-authored rubrics. Our findings highlight the importance of iterative rubric refinement in LLM-based AES to enhance alignment with human evaluations.