This paper introduces the generalized exponential- $U$ family of distributions as a novel methodological approach to enhance the distributional flexibility of existing classical and modified distributions. The new family is derived by combining the T- $X$ family method with the exponential model. The paper presents the generalized exponential-Weibull model, an updated version of the Weibull model. Estimators and heavy-tailed characteristics of the proposed method are derived. The new model is applied to three healthcare data sets, including COVID-19 patient survival times and mortality rate data set from Mexico and Holland. The proposed model outperforms other models in terms of analyzing healthcare data sets by evaluating the best model selection measures. The findings suggest that the proposed model holds promise for broader utilization in the area of predicting and modeling healthcare phenomena.