This paper examines the estimation of model parameters, reliability, and hazard rate functions of the inverted exponentiated Rayleigh distribution under progressive hybrid Type-I censoring. Three estimation methodologies, maximum likelihood, maximum product of spacing, and Bayesian approaches, are explored. The classical perspective employs maximum likelihood and maximum product of spacing approaches for estimating unknown parameters, reliability, hazard rate functions, and computing approximate confidence intervals. Bayesian estimation is formulated using the squared-error and LINEX loss functions, predicated on independent gamma priors. Owing to the complex nature of the joint posterior distribution, Bayes estimates are evaluated by generating samples from the whole conditional distributions via Markov Chain Monte Carlo methods. The highest posterior density credible intervals are also established for each unknown parameter, reliability, and hazard rate functions. The efficacy of the proposed strategies is evaluated through a simulated study. To assess the efficacy of the estimation techniques, a comprehensive simulation study is conducted, encompassing various scenarios with diverse sample sizes and progressive censoring schemes. Furthermore, the practical applicability of the proposed methods is demonstrated through the analysis of real-world datasets taken from the medical field. This data represents the relief time (in hours) of arthritis patients receiving a fixed dose of this drug. Numerical investigations reveal that Bayes estimates employing the LINEX loss function exhibit superior performance compared to other estimation methods, underscoring their preference due to heightened accuracy and robustness.