This paper extends the results previously presented by Leiva and coauthors [11] and therefore the new distribution family offers a larger set of models. Considering the Leiva et al. proposal as a standard alternative, the new family can fit data for which their proposal might be not flexible enough. As expected, in all examples, the Kolmogorov–Smirnov tests indicated a closer fit of the data to the gpower-normal distribution in comparison with the glog-normal fit.