International Classification of Diseases (ICD) coding assigns diagnosis codes to clinical documents and is essential for healthcare billing and clinical analysis. Reliable coding requires that each predicted code be supported by explicit textual evidence. However, existing public datasets provide only code labels, without evidence annotations, limiting models' ability to learn evidence-grounded predictions. In this work, we argue that dense, document-level evidence annotation is not always necessary for learning evidence-based coding. Instead, models can learn code-specific evidence patterns from local spans and use these patterns to support document-level evidence-based coding. Based on this insight, we propose Span-Centric Learning (SCL), a training framework that strengthens LLMs' coding ability at the span level and transfers this capability to full clinical documents. Specifically, we use a small set of annotated documents to supervise evidence recognition, aggregation, and code assignment, while leveraging a large collection of lightweight evidence spans to reinforce span-level reasoning. Due to their compactness, span annotations are scalable and can be further augmented through synthesis. Under the same Llama3.1-8B backbone, our approach achieves an 8.2-point improvement in macro-F1 at only 20% of the training cost of standard SFT, and provides explicit supporting evidence for each predicted code, enabling human auditing and revision.