The Weibull distribution is a cornerstone in reliability analysis. In this study, we propose a new family of distributions derived from the Weibull model, called the new Weibull-H (NW-H) family. This family features sub-models that can represent diverse failure rate patterns, including bathtub, increasing, unimodal, decreasing, J-shaped, and inverted J-shaped shapes. Additionally, these sub-models produce various density shapes, such as left-skewed, unimodal, symmetric, bimodal, right-skewed, and J-shaped. We explore the mathematical properties of the NW-H family and focus on the parameters of a specific submodel, the NW-exponential (NWEx), employing ten different estimation techniques, both classical and Bayesian. Bayesian estimators are calculated using three distinct loss functions. Through numerical simulations, we compare and rank these estimation methods based on partial and overall performance. Our results demonstrate that the Bayesian approach consistently yields the most effective parameter estimates for the NWEx across various loss functions. Moreover, we apply the NWEx distribution to three real-world datasets from engineering and industry, showcasing its superior fit and flexibility compared to several competing exponential models.