Out-of-Distribution (OOD) detection in semantic segmentation aims to localize anomalous regions at the pixel level, advancing beyond traditional image-level OOD techniques to better suit real-world applications such as autonomous driving. Recent literature has successfully explored the adaptation of commonly used image-level OOD methods--primarily based on classifier-derived confidence scores (e.g., energy or entropy)--for this pixel-precise task. However, these methods inherit a set of limitations, including vulnerability to overconfidence. In this work, we introduce SupLID, a novel framework that effectively guides classifier-derived OOD scores by exploiting the geometrical structure of the underlying semantic space, particularly using Linear Intrinsic Dimensionality (LID). While LID effectively characterizes the local structure of high-dimensional data by analyzing distance distributions, its direct application at the pixel level remains challenging. To overcome this, SupLID constructs a geometrical coreset that captures the intrinsic structure of the in-distribution (ID) subspace. It then computes OOD scores at the superpixel level, enabling both efficient real-time inference and improved spatial smoothness. We demonstrate that geometrical cues derived from SupLID serve as a complementary signal to traditional classifier confidence, enhancing the model's ability to detect diverse OOD scenarios. Designed as a post-hoc scoring method, SupLID can be seamlessly integrated with any semantic segmentation classifier at deployment time. Our results demonstrate that SupLID significantly enhances existing classifier-based OOD scores, achieving state-of-the-art performance across key evaluation metrics, including AUR, FPR, and AUP. Code is available at https://github.com/hdnugit/SupLID.