This study aims to evaluate the melting characteristics of a phase change material (PCM) in a latent heat storage system equipped with hemispherical and quarter-spherical fins. A vertical triple-pipe heat exchanger is used as the PCM-based heat storage unit to improve the melting performance compared with a double-pipe system. Furthermore, the fins are arranged in inline and staggered configurations to improve heat transfer performance. For the quarter-spherical fins, both upward and downward directions are examined. The results of the system equipped with novel fins are compared with those without fins. Moreover, a fin is added to the heat exchanger’s base to compensate for the natural convection effect at the bottom of the heat exchanger. Considering similar fin volumes, the results show that the system equipped with four hemispherical fins on the side walls and an added fin on the bottom wall has the best performance compared with the other cases with hemispherical fins. The staggered arrangement of the fins results in a higher heat transfer rate. The downward quarter-spherical fins with a staggered configuration show the highest performance among all the studied cases. Compared with the case without fins, the heat storage rate improves by almost 78% (from 35.6 to 63.5 W), reducing the melting time by 45%.