The Service Network Design Problem (SNDP) is generally considered as a fundamental problem in transportation logistics and involves the determination of an efficient transportation network and corresponding schedules. The problem is extremely challenging due to the complexity of the constraints and the scale of real-world applications. Therefore, efficient solution methods for this problem are one of the most important research issues in this field. However, current research has mainly focused on various sophisticated high-level search strategies in the form of different local search metaheuristics and their hybrids. Little attention has been paid to novel neighbourhood structures which also play a crucial role in the performance of the algorithm. In this research, we propose a new efficient neighbourhood structure that uses the SNDP constraints to its advantage and more importantly appears to have better reachability than the current ones. The effectiveness of this new neighbourhood is evaluated in a basic Tabu Search (TS) metaheuristic and a basic Guided Local Search (GLS) method. Experimental results based on a set of well-known benchmark instances show that the new neighbourhood performs better than the previous arc-flipping neighbourhood. The performance of the TS metaheuristic based on the proposed neighbourhood is further enhanced through fast neighbourhood search heuristics and hybridisation with other approaches.