Metal hydride is a type of solid hydrogen storage material that offers excellent hydrogen absorption and desorption reaction properties. However, there is a strong thermal effect during the reaction process when the materials are applied in reactors. Thus, effective thermal management techniques are crucial for promoting efficient reactions. Prior studies on the subject have shown that phase change materials can be coupled with metal hydride reactors to transfer the reaction heat. To further enhance the transfer of heat and reaction efficiency of the metal hydride reactors based on phase change materials, this study explored heat transfer enhancement methods on the phase change materials side, including adding heat transfer fins and discussing the fin parameters and composite foam metal. The findings of the study showed that in comparison to the reactor without fins, the absorption rate of adding 10 sets of fins can increase by 28%, and as the number of fins increased to 14, this value could increase to 39.4%. The use of Y-shaped fins did not substantially improve the reaction rate, mainly due to the sufficient number of straight fins, and the heat transfer area reaching the limit of the fin enhanced the process of heat transport. Thus, the improved method of using copper foam is further considered. Compared with the reactor with and without fins, the absorption rate of the copper foam coupled reactor is increased by 23.4% and 61.3%, respectively.