Non-independent and identically distributed (Non-IID) data across edge clients have long posed significant challenges to federated learning (FL) training in edge computing environments. Prior works have proposed various methods to mitigate this statistical heterogeneity. While these works can achieve good theoretical performance, in this work we provide the first investigation into a hidden over-correction phenomenon brought by the uniform model correction coefficients across clients adopted by existing methods. Such over-correction could degrade model performance and even cause failures in model convergence. To address this, we propose TACO, a novel algorithm that addresses the non-IID nature of clients' data by implementing fine-grained, client-specific gradient correction and model aggregation, steering local models towards a more accurate global optimum. Moreover, we verify that leading FL algorithms generally have better model accuracy in terms of communication rounds rather than wall-clock time, resulting from their extra computation overhead imposed on clients. To enhance the training efficiency, TACO deploys a lightweight model correction and tailored aggregation approach that requires minimum computation overhead and no extra information beyond the synchronized model parameters. To validate TACO's effectiveness, we present the first FL convergence analysis that reveals the root cause of over-correction. Extensive experiments across various datasets confirm TACO's superior and stable performance in practice.