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Petabytes of data are generated each day by emerging Internet of Things (IoT), but only few of them can be finally collected and used for Machine Learning (ML) purposes due to the apprehension of data & privacy leakage, which seriously retarding ML's growth. To alleviate this problem, Federated learning is proposed to perform model training by multiple clients' combined data without the dataset sharing within the cluster. Nevertheless, federated learning introduces massive communication overhead as the synchronized data in each epoch is of the same size as the model, and thereby leading to a low communication efficiency. Consequently, variant methods mainly focusing on the communication rounds reduction and data compression are proposed to reduce the communication overhead of federated learning. In this paper, we propose Overlap-FedAvg, a framework that parallels the model training phase with model uploading & downloading phase, so that the latter phase can be totally covered by the former phase. Compared to vanilla FedAvg, Overlap-FedAvg is further developed with a hierarchical computing strategy, a data compensation mechanism and a nesterov accelerated gradients~(NAG) algorithm. Besides, Overlap-FedAvg is orthogonal to many other compression methods so that they can be applied together to maximize the utilization of the cluster. Furthermore, the theoretical analysis is provided to prove the convergence of the proposed Overlap-FedAvg framework. Extensive experiments on both conventional and recurrent tasks with multiple models and datasets also demonstrate that the proposed Overlap-FedAvg framework substantially boosts the federated learning process.