We herein propose an efficient video coding system (EVCS) consists of a conventional codec and an adaptive overfitted multi-scale attention network (MSAN) to improve coding efficiency. At the encoder side, the MSAN adjusts the network size adaptively and is trained in an overfitting way for a group of frames. Using only the current encoding video stream as a training set, the MSAN can easily obtain powerful restoration capability. After training, the learned parameters of the MSAN are transmitted to the decoder as part of the encoded bitstream. At the decoder side, the MSAN loaded with the transmitted parameters can restore the reconstructed frames very meticulously. Compared with the high efficiency video coding (HEVC) standard, the EVCS can achieve 12.141% Bjøntegaard-Delta bitrate reduction, which outperforms existing deep learning based compressed video restoration works with less computation complexity. Moreover, the MSAN is an additional part of the conventional codec without any structure change, thereby rendering it compatible with the existing coding systems.