Banyan networks and their improved switches such as 2-dilated banyan networks are usually constructed by a self-routing mechanism, and provide a high multiplexing transmission capacity to ATM networks. Due to cell blocking in the switching elements in these banyan networks, however, cell loss is occurred and then the throughput of each network is decreased. To improve this problem, we have introduced bypasses to the original and the 2-dilated banyan networks. This paper focuses on the position of the bypasses in these banyan networks and proposes the one-bypass-connection methods in order to minimize cell transfer delay caused by the bypasses. We also analyze output rate of each network and show that the bypass method gives network designers flexible selections for network performance and transfer delay.