In this paper, we develop a new mathematical strategy to create flexible lifetime distributions. This strategy is based on a special generalized mixture derived to the one involved in the so-called weighted exponential distribution. Thus, we introduce a new class of lifetime distributions called "special generalized mixture" class and discussed its qualities. In particular, a short list of new lifetime distributions is presented in details, with a focus on the one based on the Lomax distribution. Different mathematical properties are described, including distributional results, diverse moments measures, incomplete moments, characteristic function and bivariate extensions. Then, the applicability of the new class is investigated through the model parameters based on the Lomax distribution and the analysis of a practical data set.