Maximal clique enumeration (MCE) is crucial for tasks like community detection and biological network analysis. Existing algorithms typically adopt the branch-and-bound framework with the vertex-oriented Bron-Kerbosch (BK) branching strategy, which forms the sub-branches by expanding the partial clique with a vertex. In this paper, we present a novel approach called HBBMC, a hybrid framework combining vertex-oriented BK branching and edge-oriented BK branching, where the latter adopts a branch-and-bound framework which forms the sub-branches by expanding the partial clique with an edge. This hybrid strategy enables more effective pruning and helps achieve a worst-case time complexity better than the best known one under a condition that holds for the majority of real-world graphs. To further enhance efficiency, we introduce an early termination technique, which leverages the topological information of the graphs and constructs the maximal cliques directly without branching. Our early termination technique is applicable to all branch-and-bound frameworks. Extensive experiments demonstrate the superior performance of our techniques.