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Developing open-set classification methods capable of classifying in-distribution (ID) data while detecting out-of-distribution (OOD) samples is essential for deploying graph neural networks (GNNs) in open-world scenarios. Existing methods typically treat all OOD samples as a single class, despite real-world applications—especially high-stake settings like fraud detection and medical diagnosis—demanding deeper insights into OOD samples, including their probable labels. This raises a critical question: Can OOD detection be extended to OOD classification without true label information? To answer this question, we introduce a Coarse-to-Fine open-set Classification (CFC) method that leverages large language models (LLMs) for text-attributed graphs. CFC consists of three key components: (1) A coarse classifier that utilizes LLM prompts for OOD detection and outlier label generation; (2) A GNN-based fine classifier trained with OOD samples from (1) for enhanced OOD detection and ID classification; and (3) Refined OOD classification achieved through LLM prompts and post-processed OOD labels. Unlike methods relying on synthetic or auxiliary OOD samples, CFC employs semantic OOD data-instances that are genuinely out-of-distribution based on their inherent meaning, thus improving interpretability and practical utility. CFC enhances OOD detection by 10% compared to state-of-the-art approaches on text-attributed graphs and in the text domain, while achieving up to 70% accuracy in OOD classification on graph datasets.