A novel extended Lindley lifetime model that exhibits unimodal or decreasing density shapes as well as increasing, bathtub or unimodal-then-bathtub failure rates, named the Marshall-Olkin-Lindley (MOL) model is studied. In this research, using a progressive Type-II censored, various inferences of the MOL model parameters of life are introduced. Utilizing the maximum likelihood method as a classical approach, the estimators of the model parameters and various reliability measures are investigated. Against both symmetric and asymmetric loss functions, the Bayesian estimates are obtained using the Markov Chain Monte Carlo (MCMC) technique with the assumption of independent gamma priors. From the Fisher information data and the simulated Markovian chains, the approximate asymptotic interval and the highest posterior density interval, respectively, of each unknown parameter are calculated. Via an extensive simulated study, the usefulness of the various suggested strategies is assessed with respect to some evaluation metrics such as mean squared errors, mean relative absolute biases, average confidence lengths, and coverage percentages. Comparing the Bayesian estimations based on the asymmetric loss function to the traditional technique or the symmetric loss function-based Bayesian estimations, the analysis demonstrates that asymmetric loss function-based Bayesian estimations are preferred. Finally, two data sets, representing vinyl chloride and repairable mechanical equipment items, have been investigated to support the approaches proposed and show the superiority of the proposed model compared to the other fourteen lifetime models.