In the era of the sharing economy, ridesharing has emerged as an efficient and environmentally friendly transportation mode, with route recommendation playing a vital role. Traditional methods primarily rely on shortest-path strategies for route planning, which often overlook opportunities for drivers to slightly extend their routes to capture additional requests. To address this limitation, we introduce the Demand-aware Order Dispatch with Route Recommendation (DODRE) problem, aiming to recommend routes that pass through high-demand areas and improve the overall request fulfillment rate. To solve this problem, we propose a Regional Demand-aware Route Search method, which identifies routes with the highest demand potential by considering future demand distributions across regions. This demand-aware route planning is integrated with an insertion-based allocation strategy, ensuring efficient and effective order dispatch. Extensive experiments on real-world datasets demonstrate that our approach significantly enhances service rates and outperforms traditional shortest-path-based methods in both effectiveness and computational efficiency, highlighting its potential to optimize resource utilization in ridesharing systems.