Transportation is vital in healthcare and can save lives by allowing patients to receive medical services on time, especially those with serious conditions or mobility issues. Effective transportation networks contribute to better health outcomes and help avoid treatment delays. This paper addresses the Multi-Trip Dial-a-Ride Problem (MTDARP) for non-emergency patient transport, which is essential for enabling patients to attend therapy sessions, specialized treatments, and routine check-ups. In this context, ambulances must plan multi-stop routes while respecting time windows, vehicle capacity limits, and precedence constraints. Each route begins and ends at the depot (hospital), with ambulances picking up patients at their origins and delivering them to their destinations. We propose a new Variable Neighborhood Search (VNS) algorithm to optimize routes, leveraging systematic neighborhood exploration to escape local optima. Using real-world data from the Hong Kong Hospital Authority, we demonstrate that VNS achieves solutions up to 28% better than a state-of-the-art Memetic Algorithm (MA), while reducing computational time by 90%. Our approach provides hospitals with a practical, scalable, and cost-effective scheduling tool.