Reliability evaluation holds significant importance in multiple fields, especially in engineering. Given the fast-paced advancement of modern products, researchers face the challenge of gathering a suitable amount of observed data on products that exhibit high reliability. An adaptive progressively Type-II hybrid censoring strategy is a commonly used form of censorship that helps to improve the accuracy of statistical tests by ending the experiment after getting a predetermined number of observed data. In this paper, we use this technique when the parent distribution of the population under consideration is the exponentiated Teissier distribution. We use the likelihood method to calculate point and interval estimates for model parameters and reliability indices. To determine the required interval ranges for various parameters, we use both the normal approximation of likelihood estimates and the normal approximation of their logarithm. Additionally, the Bayesian estimation method is employed to obtain point estimates and two types of credible intervals by sampling from the full conditional distributions. A simulation experiment is carried out to compare different approaches through varied experimental plans, effective number of failures, and priors. Two engineering applications are considered by analyzing the failure times of electronic components and aircraft windshields.