Container migration has emerged as the most effective way to ensure the proximity of time-critical microservices at the network edge with mobile end devices. However, ensuring service continuity while migrating microservices that rely on an established TCP connection is still a significant technical challenge. In this paper, we investigate such pivotal issue and propose COAT, a novel, yet simple, network architecture that leverages overlay network technology to achieve TCP connection migration. Through experimental validation using sample microservices, we show that, compared to the traditional container migration approach that does not support connection migration, our solution enables the successful migration of microservices relying on an established TCP connection, at the cost of a 14% maximum increase of the migration duration. Importantly, our solution to the problem of connection migration does not require the use of a dedicated protocol, or any modification to the application source code or the kernel.