With the complexity of constrained multiobjective problems (CMOPs), constrained multiobjective evolutionary algorithms (CMOEAs) are gradually becoming an excellent method to deal with CMOPs. However, with the strictness of the constraints, CMOEAs often fail to find the fully constrained Pareto Front (CPF) when optimizing CMOPs. Existing CMOEAs mostly perform less well when faced with this situation. To improve such problems, current algorithms mostly enhance the population diversity or temporarily ignore constraints hope that the population is widely distributed in the objective space to explore the complete CPF. Few scholars have taken the approach of generating elite solutions to such problems. This article proposes a constrained optimization algorithm named EGCMO. It first explores all possible regions of CPFs and then makes the elite solution to generate offspring. Specifically, in the early stage, the algorithm explores as many feasible regions as possible, maintaining the possibility that the population generates all feasible solutions. In the later stage, the population focuses on the generation of elite solutions, and the superior feasible solutions generate new offspring to provide good convergence for the population. In addition, a diversity maintenance strategy is employed to prevent the algorithm from falling into a local optimum at a later stage. Finally, EGCMO experimented with four other excellent CMOEAs on three test suites. The experimental results demonstrated that EGCMO has strong competitiveness and can handle CMOEAs more effectively.