This work introduces a deterministic-to-exploratory enhanced posterior for data assimilation (DEEP-DA), a framework that leverages Bayesian inference techniques for the estimation of unknown physical parameters and evolution of complex physical states modeled by partial differential equations, particularly in high-speed gas dynamics where predictive uncertainty is large due to shock dynamics. The DEEP-DA approach to parameter estimation integrates traditional DA techniques with Markov chain Monte Carlo (MCMC) strategies to fully capture uncertainty while also limiting computational cost. The novelty of DEEP-DA stems from its hybrid Bayesian structure, which couples the maximum likelihood ensemble filter (MLEF) with transport-map-accelerated MCMC (TM-MCMC). MLEF provides rapid estimation of both the system state and uncertain parameters by solving an optimization problem at each assimilation cycle. The MLEF analysis is then used to construct transport maps that accelerate MCMC sampling, an approach for generating random samples from complex distributions, leading to efficient and accurate exploration of non-Gaussian and nonlinear posterior parameter distributions. Traditional DA methods, such as MLEF, are computationally efficient but rely on Gaussian error assumptions and struggle with strong nonlinearities. In contrast, sampling methods like TM-MCMC provide rigorous uncertainty quantification but are expensive for the full exploration of the model posterior. The goal of DEEP-DA is to address these gaps by combining the strengths of both approaches, the posterior accuracy of TM-MCMC and the efficiency of MLEF. To assess the performance of this algorithm, DEEP-DA is applied to a one-dimensional shock tube problem where the heat capacity ratio will be updated within regions of high discontinuity. DEEP-DA provides a potential pathway toward higher-dimensional and more complex applications, such as hypersonic vehicle re-entry, where accurately characterizing nonlinear flow regimes with large uncertainty remains a challenge.