The accelerated bridge construction (ABC) method has become increasingly popular in recent decades because it creates few traffic disruptions, minimal environmental impacts, and lower cycle costs. The self-propelled modular transporter (SPMT) is an ABC method. In the SPMT method, the support position is a key factor during transportation because the support position changes the boundary conditions and consequently changes the stress distribution of the original bridge. Unreasonable stress distribution may cause a collapse of the demolished bridge and permanent damage to the new built bridge during the transportation process. In the present study, an overpass bridge is demolished and replaced due to the expansion of an expressway. The SPMT method is adopted for rapid demolition and construction because it can greatly reduce traffic interruptions and costs. An optimization strategy of SPMT support positions is proposed by introducing a safety factor. The influence of the support positions of the SPMT on the internal force, stress, deflection, and resistance to overturning of both the demolished and new built bridges during the transport process is considered in the safety factor. Finally, an overpass bridge is taken as a case study. The optimal support positions of both demolished and new built bridges are obtained, and the demolition and rebuilding of the bridge are successfully completed using the SPMT method.