Recently, new versions of the TCP protocols such as HSTCP (High Speed TCP) has proposed to adopt the rapid growth of the Internet. However, it is important to consider a deployment scenario to shift gradually from the current to the new versions of TCP. A fairness problem between the performance of both current and new versions of TCP will be arisen when they share the same bottleneck link, which leads some complaints from people using the current version of TCP. In this paper, we first clarify the relation between the deployment share of HSTCP and the negative impact to TCP Reno users. We then propose a new management mechanism which supports TCP version migration from TCP Reno to HSTCP based on the relation. Through simulation, we also show that our mechanism enables to accelerate a gradual migration of TCP versions by achieving the end users' complaints as little as possible.