Recently, the Internet is widely used as a social basis and is not only providing many kinds of services but also offering these at high speed with high reliability. As one method for such a demand, multihoming technique, by which the inside network is connected with two or more ISPs (Internet Service Providers) and can use them properly according to network condition, attracts attention. In IPv6 site multihoming environment, multiple IP addresses are usually assigned to each host in the site. However, when a packet is sent from an inside node to outside, the packet has to go through a proper site-exit router in order to avoid ingress filtering. Therefore, especially, when communicating on the outbound connection initiated on an inside node, it is impossible to select a proper site-exit router since the source IP address is selected before initiating a connection. To solve this problem, we propose a route selection method for outbound connections. This method introduces a middleware into each inside node to establish a connection via each site-exit router simultaneously and then uses the first established connection. This middleware also introduces a kind of Network Address Translation (NAT) function to avoid ingress filtering. According to simulation experiments, we confirmed that the proposed method can select proper routes.