Auditing machine learning-based (ML) healthcare tools for bias is critical to preventing patient harm, especially in communities that disproportionately face health inequities. General frameworks are becoming increasingly available to measure ML fairness gaps between groups. However, ML for health (ML4H) auditing principles call for a contextual, patient-centered approach to model assessment. Therefore, ML auditing tools must be (1) better aligned with ML4H auditing principles and (2) able to illuminate and characterize communities vulnerable to the most harm. To address this gap, we propose supplementing ML4H auditing frameworks with SLOGAN (patient Severity-based LOcal Group biAs detectioN), an automatic tool for capturing local biases in a clinical prediction task. SLOGAN adapts an existing tool, LOGAN (LOcal Group biAs detectioN), by contextualizing group bias detection in patient illness severity and past medical history. We investigate and compare SLOGAN's bias detection capabilities to LOGAN and other clustering techniques across patient subgroups in the MIMIC-III dataset. On average, SLOGAN identifies larger fairness disparities in over 75% of patient groups than LOGAN while maintaining clustering quality. Furthermore, in a diabetes case study, health disparity literature corroborates the characterizations of the most biased clusters identified by SLOGAN. Our results contribute to the broader discussion of how machine learning biases may perpetuate existing healthcare disparities.