Pre-training large language models (LLMs) on vast text corpora enhances natural language processing capabilities but risks encoding social biases, particularly gender bias. While parameter-modification methods like fine-tuning mitigate bias, they are resource-intensive, unsuitable for closed-source models, and lack adaptability to evolving societal norms. Instruction-based approaches offer flexibility but often compromise general performance on normal tasks. To address these limitations, we propose $\textit{FaIRMaker}$, an automated and model-independent framework that employs an $\textbf{auto-search and refinement}$ paradigm to adaptively generate Fairwords, which act as instructions to reduce gender bias and enhance response quality. $\textit{FaIRMaker}$ enhances the debiasing capacity by enlarging the Fairwords search space while preserving the utility and making it applicable to closed-source models by training a sequence-to-sequence model that adaptively refines Fairwords into effective debiasing instructions when facing gender-related queries and performance-boosting prompts for neutral inputs. Extensive experiments demonstrate that $\textit{FaIRMaker}$ effectively mitigates gender bias while preserving task integrity and ensuring compatibility with both open- and closed-source LLMs.