Out-of-Distribution (OOD) detection requires sensitivity to subtle shifts without overreacting to natural In-Distribution (ID) diversity. However, from the viewpoint of detection granularity, global representation inevitably suppress local OOD cues, while patch-based methods are unstable due to entangled spurious-correlation and noise. And neither them is effective in detecting compositional OODs composed of valid ID components. Inspired by recognition-by-components theory, we present a training-free Component-Based OOD Detection (CoOD) framework that addresses the existing limitations by decomposing inputs into functional components. To instantiate CoOD, we derive Component Shift Score (CSS) to detect local appearance shifts, and Compositional Consistency Score (CCS) to identify cross-component compositional inconsistencies. Empirically, CoOD achieves consistent improvements on both coarse- and fine-grained OOD detection.