Detecting out-of-distribution (OOD) samples is crucial for ensuring the safety and robustness of models deployed in real-world scenarios. While most studies on OOD detection focus on fine-tuned models trained on in-distribution (ID) data, detecting OOD in pre-trained models is also important due to computational limitations and the widespread use of open-source pre-trained models. However, in the same domain shift setting, the OOD detection performance of pre-trained models is insufficient because both ID and OOD samples originate from the same domain, leading to a high overlap in their embeddings. To address this issue, we introduce a new method called CED, a training-free OOD detection technique designed to enhance the distinction between ID and OOD datasets. We theoretically validate that specific auxiliary and oracle samples that satisfy certain conditions improve this distinction. Motivated by our theoretical analysis, CED enhances the differentiation by utilizing these specially designed auxiliary and oracle samples. As a result, CED significantly improves the ability of pre-trained models to distinguish between ID and OOD samples in text classification and hallucination detection tasks. Furthermore, we verify that CED is a plug-and-play method compatible with various backbone networks, such as RoBERTa, Llama, and OpenAI Embedding.