Employing phase-change materials (PCM) is considered a very efficient and cost-effective option for addressing the mismatch between the energy supply and the demand. The high storage density, little temperature degradation, and ease of material processing register the PCM as a key candidate for the thermal energy storage system. However, the sluggish response rates during their melting and solidification processes limit their applications and consequently require the inclusion of heat transfer enhancers. This research aims to investigate the potential enhancement of circular fins on intensifying the PCM thermal response in a vertical triple-tube casing. Fin arrays of non-uniform dimensions and distinct distribution patterns were designed and investigated to determine the impact of modifying the fin geometric characteristics and distribution patterns in various spatial zones of the heat exchanger. Parametric analysis on the various fin structures under consideration was carried out to determine the most optimal fin structure from the perspective of the transient melting evolution and heat storage rates while maintaining the same design limitations of fin material and volume usage. The results revealed that changing the fin dimensions with the heat-flow direction results in a faster charging rate, a higher storage rate, and a more uniform temperature distribution when compared to a uniform fin size. The time required to fully charge the storage system (fully melting of the PCM) was found to be reduced by up to 10.4%, and the heat storage rate can be improved by up to 9.3% compared to the reference case of uniform fin sizes within the same fin volume limitations.