Score function-based natural language generation (NLG) approaches such as REINFORCE, in general, suffer from low sample efficiency and training instability problems. This is mainly due to the non-differentiable nature of the discrete space sampling and thus these methods have to treat the discriminator as a black box and ignore the gradient information. To improve the sample efficiency and reduce the variance of REINFORCE, we propose a novel approach, TaylorGAN, which augments the gradient estimation by off-policy update and the first-order Taylor expansion. This approach enables us to train NLG models from scratch with smaller batch size --- without maximum likelihood pre-training, and outperforms existing GAN-based methods on multiple metrics of quality and diversity.