Large Language Models (LLMs) have gained significant traction in software engineering for automating tasks such as unit test generation. Most existing studies prioritize code coverage as the primary metric for enhancing test suite effectiveness. However, prior research has shown that although code coverage can reach approximately 80%, the mutation score, which generally exhibits a stronger correlation with defect detection effectiveness, attains only about 35%. This gap highlights the need to enhance test suite effectiveness guided by mutation score rather than code coverage. Recent studies, including MuTAP and MutGen, explored the use of survived mutants to enhance test suite effectiveness. However, their evaluations were limited to simple standalone methods that rely on built-in functions and standard libraries. Non-standalone methods, which depend on other classes and involve complex user-defined types, are more intricate and commonly found in real-world projects. The limited contextual information and basic repair mechanisms in their prompt designs make it unclear whether their performance can generalize to non-standalone methods. Moreover, the two studies rely on existing language-specific, rule-based mutation techniques, which require specific configurations and incur additional costs when adapting to other programming languages. To bridge this gap, we propose a novel, fully automatic LLM-based approach to enhance test suite effectiveness, guided by survived mutants. The approach augments initial test suites by integrating mutation testing with test case generation. It takes focal method information as input and generates test cases targeting survived mutants identified from applying the initial test suites. Our approach incorporates multiple prompt techniques, rich contextual information, and an advanced repair mechanism to effectively generate test cases for non-standalone methods. The evaluation covers 1,035 focal methods, categorized as standalone or non-standalone. On average, the mutation score increases by 16.11% for standalone methods and 8.09% for non-standalone methods. We validate the practical impact of augmented test suites in LLM-based code generation. After test suite augmentation, pass@1 decreased by 0.3152 and 0.1772 on average for standalone and non-standalone methods, respectively, indicating the effectiveness of our approach in reducing false positives caused by insufficient test cases in code generation evaluation.