Due to the potential cost saving and minimal temperature stratification, the energy storage based on phase-change materials (PCMs) can be a reliable approach for decoupling energy demand from immediate supply availability. However, due to their high heat resistance, these materials necessitate the introduction of enhancing additives, such as expanded surfaces and fins, to enable their deployment in more widespread thermal and energy storage applications. This study reports on how circular fins with staggered distribution and variable orientations can be employed for addressing the low thermal response rates in a PCM (Paraffin RT-35) triple-tube heat exchanger consisting of two heat-transfer fluids flow in opposites directions through the inner and the outer tubes. Various configurations, dimensions, and orientations of the circular fins at different flow conditions of the heat-transfer fluid were numerically examined and optimized using an experimentally validated computational fluid-dynamic model. The results show that the melting rate, compared with the base case of finless, can be improved by 88% and the heat charging rate by 34%, when the fin orientation is downward–upward along the left side and the right side of the PCM shell. The results also show that there is a benefit if longer fins with smaller thicknesses are adopted in the vertical direction of the storage unit. This benefit helps natural convection to play a greater role, resulting in higher melting rates. Changing the fins’ dimensions from (thickness × length) 2 × 7.071 mm2 to 0.55 × 25.76 mm2 decreases the melting time by 22% and increases the heat charging rate by 9.6%. This study has also confirmed the importance of selecting the suitable values of Reynolds numbers and the inlet temperatures of the heat-transfer fluid for optimizing the melting enhancement potential of circular fins with downward–upward fin orientations.