If a shared IP network is to deliver large-volume streaming media content, such as real-time videos, we need a technique for explicitly setting and dynamically changing the transmission paths used to respond to the congestion situation of the network, including multi-path transmission of a single-flow, to maximize network bandwidth utilization and stabilize transmission quality. However, current technologies cannot realize flexible multi-path transmission because they require complicated algorithms for route searching and the control load for route changing is excessive. This paper proposes a scheme that realizes routing control for multi-path transmission by combining multiple virtual networks on the same physical network. The proposed scheme lowers the control load incurred in creating a detour route because routing control is performed by combining existing routing planes. In addition, our scheme simplifies route searching procedure because congestion avoidance control of multi-path transmission can be realized by the control of a single path. An experiment on the JGN-X network virtualization platform finds that while the time taken to build an inter-slice link must be improved, the time required to inspect whether each slice has virtual nodes that can be connected to the original slice and be used as a detour destination can be as short as 40 microseconds per slice even with large slices having more than 100 virtual nodes.