A new one-parameter Chris-Jerry distribution, created by mixing exponential and gamma distributions, is discussed in this article in the presence of incomplete lifetime data. We examine a novel generalized progressively hybrid censoring technique that ensures the experiment ends at a predefined period when the model of the test participants has a Chris-Jerry (CJ) distribution. When the indicated censored data is present, Bayes and likelihood estimations are used to explore the CJ parameter and reliability indices, including the hazard rate and reliability functions. We acquire the estimated asymptotic and credible confidence intervals of each unknown quantity. Additionally, via the squared-error loss, the Bayes’ estimators are obtained using gamma prior. The Bayes estimators cannot be expressed theoretically since the likelihood density is created in a complex manner; nonetheless, Markov-chain Monte Carlo techniques can be used to evaluate them. The effectiveness of the investigated estimations is assessed, and some recommendations are given using Monte Carlo results. Ultimately, an analysis of two engineering applications, such as mechanical equipment and ball bearing data sets, shows the applicability of the proposed approaches that may be used in real-world settings.