The Gaussian distribution is widely used in mechanism design for differential privacy (DP). Thanks to its sub-Gaussian tail, it significantly reduces the chance of outliers when responding to queries. However, it can only provide approximate DP. In this paper, we introduce and analyse a new distribution for use in DP that is based on the Gaussian distribution, but it has improved privacy performance. The so-called offset-symmetric Gaussian tail (OSGT) distribution is obtained through using the normalized tails of two symmetric Gaussians around zero. Consequently, it can still have sub-Gaussian tail and lend itself to analytical derivations. We analytically derive the variance of the OSGT random variable and its differential privacy metrics. Numerical results show the OSGT mechanism can offer better privacy-utility performance compared to the Gaussian and Laplace mechanisms.