A reliability experiment is a crucial determinant of a component's success, as it simulates the effects of aging and usage by exposing the component in a system to higher levels of stress or wear. Consequently, assessments of the component's performance and its capacity to satisfy consumers are conducted. This study aims to estimate the stress-strength inequality function R = P(Y < X) after subjecting it to a step-stress partially accelerated life test, which acts as a secondary test by exposing the component to more extreme conditions than usual. The strength variable (X) and stress variable (Y) are assumed to be independent random variables following the Gompertz distribution. The maximum likelihood method was used to estimate the model parameters and the stress-strength parameter \textit{R}. Two alternative estimation techniques were utilized, specifically the Bayesian method proposing gamma priors for the model's parameters and E-Bayesian estimation suggesting beta priors for the hyperparameters. Furthermore, Lindley's approximation method and Markov Chain Monte Carlo simulation were utilized with both the squared error and the precautionary loss functions to derive Bayesian and E-Bayesian estimators. Interval estimation methods such as asymptotic confidence intervals, Bayes, and expected Bayes credible intervals were discussed. A simulation study and real-data application were employed to evaluate the proposed estimating methods that have been developed in addition to verifying the accuracy of the results.discussed. A simulation study and real-data application were employed to evaluate the proposed estimating methods that have been developed in addition to verifying the accuracy of the results.