With the rise of computing power, using data-driven approaches for co-designing robots' morphology and controller has become a feasible way. Nevertheless, evaluating the fitness of the controller under each morphology is time-consuming. As a pioneering data-driven method, Co-adaptation utilizes a double-network mechanism with the aim of learning a Q function conditioned on morphology parameters to replace the traditional evaluation of a diverse set of candidates, thereby speeding up optimization. In this paper, we find that Co-adaptation ignores the existence of exploration error during training and state-action distribution shift during parameter transmitting, which hurt the performance. We propose the framework of the concurrent network that couples online and offline RL methods. By leveraging the behavior cloning term flexibly, we mitigate the impact of the above issues on the results. Simulation and physical experiments are performed to demonstrate that our proposed method outperforms baseline algorithms, which illustrates that the proposed method is an effective way of discovering the optimal combination of morphology and controller.