Inherent fluctuations in the availability of energy from renewables, particularly solar, remain a substantial impediment to their widespread deployment worldwide. Employing phase-change materials (PCMs) as media, saving energy for later consumption, offers a promising solution for overcoming the problem. However, the heat conductivities of most PCMs are limited, which severely limits the energy storage potential of these materials. This study suggests employing circular fins with staggered distribution to achieve improved thermal response rates of PCM in a vertical triple-tube heat exchanger involving two opposite flow streams of the heat-transfer fluid (HTF). Since heat diffusion is not the same at various portions of the PCM unit, different fin configurations, fin dimensions and HTF flow boundary conditions were explored using computational studies of melting in the PCM triple-tube system. Staggered configuration of fin distribution resulted in significant increases in the rates of PCM melting. The results indicate that the melting rate and heat charging rate could be increased by 37.2 and 59.1%, respectively, in the case of staggered distribution. Furthermore, the use of lengthy fins with smaller thickness in the vertical direction of the storage unit resulted in a better positive role of natural convection; thus, faster melting rates were achieved. With fin dimensions of 0.666 mm × 15 mm, the melting rate was found to be increased by 23.6%, when compared to the base case of 2 mm × 5 mm. Finally, it was confirmed that the values of the Reynolds number and inlet temperatures of the HTF had a significant impact on melting time savings when circular fins of staggered distribution were included.