This study aims to alleviate the trade-off between utility and privacy of differentially private clustering. Existing works focus on simple methods, which show poor performance for non-convex clusters. To fit complex cluster distributions, we propose sophisticated dynamical processing inspired by Morse theory, with which we hierarchically connect the Gaussian sub-clusters obtained through existing methods. Our theoretical results imply that the proposed dynamical processing introduces little to no additional privacy loss. Experiments show that our framework can improve the clustering performance of existing methods at the same privacy level.