The framework of differential privacy protects an individual's privacy while publishing query responses on congregated data. In this work, a new noise addition mechanism for differential privacy is introduced where the noise added is sampled from a hybrid density that resembles Laplace in the centre and Gaussian in the tail. With a sharper centre and light, sub-Gaussian tail, this density has the best characteristics of both distributions. We theoretically analyze the proposed mechanism, and we derive the necessary and sufficient condition in one dimension and a sufficient condition in high dimensions for the mechanism to guarantee ($ε$,$δ$)-differential privacy. Numerical simulations corroborate the efficacy of the proposed mechanism compared to other existing mechanisms in achieving a better trade-off between privacy and accuracy.