Spiking neural networks (SNNs) are considered as a potential candidate to overcome current challenges, such as the high-power consumption encountered by artificial neural networks (ANNs); however, there is still a gap between them with respect to the recognition accuracy on various tasks. A conversion strategy was, thus, introduced recently to bridge this gap by mapping a trained ANN to an SNN. However, it is still unclear that to what extent this obtained SNN can benefit both the accuracy advantage from ANN and high efficiency from the spike-based paradigm of computation. In this article, we propose two new conversion methods, namely TerMapping and AugMapping. The TerMapping is a straightforward extension of a typical threshold-balancing method with a double-threshold scheme, while the AugMapping additionally incorporates a new scheme of augmented spike that employs a spike coefficient to carry the number of typical all-or-nothing spikes occurring at a time step. We examine the performance of our methods based on the MNIST, Fashion-MNIST, and CIFAR10 data sets. The results show that the proposed double-threshold scheme can effectively improve the accuracies of the converted SNNs. More importantly, the proposed AugMapping is more advantageous for constructing accurate, fast, and efficient deep SNNs compared with other state-of-the-art approaches. Our study, therefore, provides new approaches for further integration of advanced techniques in ANNs to improve the performance of SNNs, which could be of great merit to applied developments with spike-based neuromorphic computing.