Language models trained via federated learning (FL) demonstrate impressive capabilities in handling complex tasks while protecting user privacy. Recent studies indicate that leveraging gradient information and prior knowledge can potentially reveal training samples within FL setting. However, these investigations have overlooked the potential privacy risks tied to the intrinsic architecture of the models. This paper presents a two-stage privacy attack strategy that targets the vulnerabilities in the architecture of contemporary language models, significantly enhancing attack performance by initially recovering certain feature directions as additional supervisory signals. Our comparative experiments demonstrate superior attack performance across various datasets and scenarios, highlighting the privacy leakage risk associated with the increasingly complex architectures of language models. We call for the community to recognize and address these potential privacy risks in designing large language models.