Recent advances in large language models (LLMs) have enabled promising progress in diagnosis prediction from electronic health records (EHRs). However, existing LLM-based approaches tend to overfit to historically observed diagnoses, often overlooking novel yet clinically important conditions that are critical for early intervention. To address this, we propose EviCare, an in-context reasoning framework that integrates deep model guidance into LLM-based diagnosis prediction. Rather than prompting LLMs directly with raw EHR inputs, EviCare performs (1) deep model inference for candidate selection, (2) evidential prioritization for set-based EHRs, and (3) relational evidence construction for novel diagnosis prediction. These signals are then composed into an adaptive in-context prompt to guide LLM reasoning in an accurate and interpretable manner. Extensive experiments on two real-world EHR benchmarks (MIMIC-III and MIMIC-IV) demonstrate that EviCare achieves significant performance gains, which consistently outperforms both LLM-only and deep model-only baselines by an average of 20.65\% across precision and accuracy metrics. The improvements are particularly notable in challenging novel diagnosis prediction, yielding average improvements of 30.97\%.