The key to OOD detection has two aspects: generalized feature representation and precise category description. Recently, vision-language models such as CLIP provide significant advances in both two issues, but constructing precise category descriptions is still in its infancy due to the absence of unseen categories. This work introduces two hierarchical contexts, namely perceptual context and spurious context, to carefully describe the precise category boundary through automatic prompt tuning. Specifically, perceptual contexts perceive the inter-category difference (e.g., cats vs apples) for current classification tasks, while spurious contexts further identify spurious (similar but exactly not) OOD samples for every single category (e.g., cats vs panthers, apples vs peaches). The two contexts hierarchically construct the precise description for a certain category, which is, first roughly classifying a sample to the predicted category and then delicately identifying whether it is truly an ID sample or actually OOD. Moreover, the precise descriptions for those categories within the vision-language framework present a novel application: CATegory-EXtensible OOD detection (CATEX). One can efficiently extend the set of recognizable categories by simply merging the hierarchical contexts learned under different sub-task settings. And extensive experiments are conducted to demonstrate CATEX’s effectiveness, robustness, and category-extensibility. For instance, CATEX consistently surpasses the rivals by a large margin with several protocols on the challenging ImageNet-1K dataset. In addition, we offer new insights on how to efficiently scale up the prompt engineering in vision-language models to recognize thousands of object categories, as well as how to incorporate large language models (like GPT-3) to boost zero-shot applications.