This article introduces a new family of distributions through the transformation involving tangent functions, referred to as the New Logarithmic Tangent-U family. The newly introduced family may be written as the NLT-U family of distribution. An overview of fundamental characteristics inherent to the proposed family of distributions is provided. We derived a special sub-model of the NLT-U distributions family using the Weibull distribution as a baseline reference. This special model is termed the New Logarithmic Tangent-Weibull (NLT-Wei) distribution. The hazard function of the special model is very flexible; it captures increasing, decreasing, unimodal, increasing-decreasing-decreasing, and bathtub shapes. The Maximum Likelihood estimation scheme is employed to estimate the model parameters of the proposed family. To assess the stability of the parameter estimation procedure, we perform a Monte Carlo simulation analysis. The results (numerically as well as graphically) of these simulations demonstration that biases and mean square errors decrease as sample sizes grow, even with smaller samples. We thoroughly examine the significance of the NLT-Wei distribution and illustrate its application using three real-world data sets from the engineering field. The suitability and flexibility of the NLT-Wei distribution are compared against several other distributions, including the Exponentiated Weibull, Kumaraswamy Weibull, New Generalized Logarithmic Weibull, Weibull, and Flexible Weibull distributions. The paper ends with a concluding discussion.