Accurate clinical coding is essential for healthcare documentation, billing, and decision-making. While prior work shows that off-the-shelf LLMs struggle with this task, evaluations based on exact match metrics often overlook errors where predicted codes are hierarchically close but incorrect. Our analysis reveals that such hierarchical misalignments account for a substantial portion of LLM failures. We show that lightweight interventions, including prompt engineering and small-scale fine-tuning, can improve accuracy without the computational overhead of search-based methods. To address hierarchically near-miss errors, we introduce clinical code verification as both a standalone task and a pipeline component. To mitigate the limitations in existing datasets, such as incomplete evidence and inpatient bias in MIMIC, we release an expert double-annotated benchmark of outpatient clinical notes with ICD-10 codes. Our results highlight verification as an effective and reliable step toward improving LLM-based medical coding.