© 2021 Informa UK Limited, trading as Taylor & Francis Group.It has been observed in the literature of probability theory that the existing probability distributions do not provide an adequate fit and fails to model the lifetime data having a non-monotonic hazard rate shapes. To cover this gap, researchers are working in the modifications of these distributions. In this paper, a new family of probability distributions is introduced called New Alpha Power Transformed Family (NAPT) of distributions. We derive a special case of NAPT by employing the Exponential distribution called New Alpha Power Transformed Exponential (NAPTE) distribution. NAPTE leads to a flexible fit than other distributions while modeling the lifetime data in both monotonic and non-monotonic hazard rate shapes. The paper presents a simulation study to delineate the performance of the parameters. To illustrate the usefulness of NAPTE, two real-life data sets have been used. From the results it is evident that NAPTE leads to a preferable fit than Weibull, Exponential, Rayleigh, and the Weibull Exponential distribution.