The Satisfiability (SAT) problem is a core challenge with significant applications in software engineering, including automated testing, configuration management, and program verification. This paper presents SolSearch, a novel framework that harnesses large language models (LLMs) to automatically discover and optimize SAT-solving strategies. Leveraging a curriculum-based, trial-and-error process, SolSearch enables the LLM to iteratively modify and generate SAT solver code, thereby improving solving efficiency and performance. This automated SAT solving paradigm has the advantage of being plug-and-play, allowing integration with any SAT solver and accelerating the development or design process of new SAT solvers (new methods). Our preliminary experimental results are encouraging by demonstrating that the LLM-powered paradigm not only improves state-of-the-art SAT solvers on general SAT benchmarks but also significantly enhances the performance of the widely used Z3 solver (11% on PAR-2 score). These results highlight the potential for using LLM-driven methods to advance solver adaptability and effectiveness in real-world software engineering challenges. Future research directions are discussed to further refine and validate this approach, offering a promising avenue for integrating AI with traditional software engineering tasks.