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Practical data assimilation algorithms often contain hyper-parameters, which may arise due to, for instance, the use of certain auxiliary techniques like covariance inflation and localization in an ensemble Kalman filter, the re-parameterization of certain quantities such as model and/or observation error covariance matrices, and so on. Given the richness of the established assimilation algorithms, and the abundance of the approaches through which hyper-parameters are introduced to the assimilation algorithms, one may ask whether it is possible to develop a sound and generic method to efficiently choose various types of (sometimes high-dimensional) hyper-parameters. This work aims to explore a feasible, although likely partial, answer to this question. Our main idea is built upon the notion that a data assimilation algorithm with hyper-parameters can be considered as a parametric mapping that links a set of quantities of interest (e.g., model state variables and/or parameters) to a corresponding set of predicted observations in the observation space. As such, the choice of hyper-parameters can be recast as a parameter estimation problem, in which our objective is to tune the hyper-parameters in such a way that the resulted predicted observations can match the real observations to a good extent. From this perspective, we propose a hyper-parameter estimation workflow and investigate the performance of this workflow in an ensemble Kalman filter. In a series of experiments, we observe that the proposed workflow works efficiently even in the presence of a relatively large amount (up to $10^3$) of hyper-parameters, and exhibits reasonably good and consistent performance under various conditions.