||Understanding of a hybrid percolation transitions (HPTs) induced by cluster coalescence, exhibiting a jump in the giant cluster size and a critical behavior of finite clusters, is fundamental and intriguing. Here, we uncover the underlying mechanism using the so-called restricted-random network model, in which clusters are ranked by size and partitioned into small- and large-cluster sets. As clusters are merged and their rankings are updated, they may move back and forth across the set boundary. The intervals of these crossings exhibit a self-organized critical (SOC) behavior with two power-law exponents. During this process, a bump is formed and eliminated in the cluster size distribution, characterizing the criticality of the HPT. This SOC behavior is in contrast to the critical branching process, which governs the avalanche dynamics of the HPT in the pruning process. Finally, we find that a burst of such crossing events occurs and signals the upcoming abrupt transition.