In this paper, a new transmuted general family of distributions is introduced and studied. It offers an original alternative to the former transmuted family by revisiting its mathematical structure, opening a new perspective on modeling. We derive expressions for the moments, the incomplete moments, the moment generating function, entropy measures, and order statistics as main properties. Further, we introduce a bivariate extension of the new family. Some members of the family are introduced, involving Burr, Gompertz, Weibull and gamma distributions. We discuss the different methods of estimation of the model parameters and a simulation study is performed for one of the special models to check the asymptotic behavior of the estimates under all estimation methods. Further, the interest of the family is illustrated by fitting two real-life data sets to the mentioned sub-models.