Searchable Symmetric Encryption (SSE) enables efficient search capabilities over encrypted data, allowing users to maintain privacy while utilizing cloud storage. However, SSE schemes are vulnerable to leakage attacks that exploit access patterns, search frequency, and volume information. Existing studies frequently assume that adversaries possess a substantial fraction of the encrypted dataset to mount effective inference attacks, implying there is a database leakage of such documents, thus, an assumption that may not hold in real-world scenarios. In this work, we investigate the feasibility of enhancing leakage attacks under a more realistic threat model in which adversaries have access to minimal leaked data. We propose a novel approach that leverages large language models (LLMs), specifically GPT-4 variants, to generate synthetic documents that statistically and semantically resemble the real-world dataset of Enron emails. Using the email corpus as a case study, we evaluate the effectiveness of synthetic data generated via random sampling and hierarchical clustering methods on the performance of the SAP (Search Access Pattern) keyword inference attack restricted to token volumes only. Our results demonstrate that, while the choice of LLM has limited effect, increasing dataset size and employing clustering-based generation significantly improve attack accuracy, achieving comparable performance to attacks using larger amounts of real data. We highlight the growing relevance of LLMs in adversarial contexts.