Autonomous control of double-Ackermann-steering robots is essential in agricultural applications, where robots must execute precise and complex maneuvers within a limited space. Classical methods, such as the Timed Elastic Band (TEB) planner, can address this problem, but they rely on parameter tuning, making them highly sensitive to changes in robot configuration or environment and impractical to deploy without constant recalibration. At the same time, end-to-end deep reinforcement learning (DRL) methods often fail due to unsuitable reward functions for non-holonomic constraints, resulting in sub-optimal policies and poor generalization. To address these challenges, this paper presents ManeuverNet, a DRL framework tailored for double-Ackermann systems, combining Soft Actor-Critic with CrossQ. Furthermore, ManeuverNet introduces four specifically designed reward functions to support maneuver learning. Unlike prior work, ManeuverNet does not depend on expert data or handcrafted guidance. We extensively evaluate ManeuverNet against both state-of-the-art DRL baselines and the TEB planner. Experimental results demonstrate that our framework substantially improves maneuverability and success rates, achieving more than a 40% gain over DRL baselines. Moreover, ManeuverNet effectively mitigates the strong parameter sensitivity observed in the TEB planner. In real-world trials, ManeuverNet achieved up to a 90% increase in maneuvering trajectory efficiency, highlighting its robustness and practical applicability.