Efficiently counting Markov equivalent directed acyclic graphs (DAGs) is crucial in graphical causal analysis. Wienöbst et al. (2023) introduced a polynomial-time algorithm, known as the Clique-Picking algorithm, to count the number of Markov equivalent DAGs for a given completed partially directed acyclic graph (CPDAG). This algorithm iteratively selects a root clique, determines fixed orientations with outgoing edges from the clique, and generates the unresolved undirected connected components (UCCGs). In this work, we propose a more efficient approach to UCCG generation by utilizing previously computed results for different root cliques. Our method introduces the concept of super cliques within rooted clique trees, enabling their efficient transfer between trees with different root cliques. The proposed algorithm effectively reduces the computational complexity of the Clique-Picking method, particularly when the number of cliques is substantially smaller than the number of vertices and edges.