A fully parametric accelerated failure time (AFT) model with a flexible, novel modified exponential Weibull baseline distribution called the extended exponential Weibull accelerated failure time (ExEW-AFT) model is proposed. The model is presented using the multi-parameter survival regression model, where more than one distributional parameter is linked to the covariates. The model formulation, probabilistic functions, and some of its sub-models were derived. The parameters of the introduced model are estimated using the maximum likelihood approach. An extensive simulation study is used to assess the estimates’ performance using different scenarios based on the baseline hazard shape. The proposed model is applied to a real-life right-censored COVID-19 data set from Sudan to illustrate the practical applicability of the proposed AFT model.