In modern applications, large graphs are usually applied in the simulation and analysis of large complex systems such as social networks, computer networks, maps, traffic networks. Therefore, graph mining is also an interesting subject attracting many researchers. Among them, frequent subgraph mining in a single large graph is one of the most important branches of graph mining, it is defined as finding all subgraphs whose occurrences in a dataset are greater than or equal to a given frequency threshold. In which, the GraMi algorithm is considered the state of the art approach and many algorithms have been proposed to improve this algorithm. In 2020, the SoGraMi algorithm was proposed to optimize the GraMi algorithm and presented an outstanding performance in terms of runtime and storage space. In this paper, we propose a new algorithm to improve SoGraMi based on connected components, called CCGraMi (Connected Components GraMi). Our experiments on four real datasets (both directed and undirected) show that the proposed algorithm outperforms SoGraMi in terms of running time as well as memory requirements.