Recent advancements in pre-trained vision-language models like CLIP have shown promise in person re-identification (ReID) applications. However, their performance in generalizable person re-identification tasks remains suboptimal. The large-scale and diverse image-text pairs used in CLIP's pre-training may lead to a lack or insufficiency of certain fine-grained features. In light of these challenges, we propose a hard sample mining method called DFGS (Depth-First Graph Sampler), based on depth-first search, designed to offer sufficiently challenging samples to enhance CLIP's ability to extract fine-grained features. DFGS can be applied to both the image encoder and the text encoder in CLIP. By leveraging the powerful cross-modal learning capabilities of CLIP, we aim to apply our DFGS method to extract challenging samples and form mini-batches with high discriminative difficulty, providing the image model with more efficient and challenging samples that are difficult to distinguish, thereby enhancing the model's ability to differentiate between individuals. Our results demonstrate significant improvements over other methods, confirming the effectiveness of DFGS in providing challenging samples that enhance CLIP's performance in generalizable person re-identification.