This paper presents TRACE-CS2, a framework for explainable course scheduling that builds on TRACE-CS. It combines a SAT-based contrastive explainer with a large language model for natural language interaction. We identify three limitations of TRACE-CS: reporting infeasibility when there exists possible schedules; accepting only one positive and one negative literal per query; and a limited schedule diversification. TRACE-CS2 addresses these limitations by incorporating: a three-step local/global feasibility check; multi-literal queries; and SAT model enumeration. On a benchmark of 600 contrastive queries, TRACE-CS2 reaches better accuracy and may produce more distinct schedules.