The significance of latent heat thermal energy storage is more substantial when compared to sensible energy storage due to its higher energy storage capability. In this paper, heat transfer enhancement techniques for melting (charging) and solidification (discharging) by using external fins and internal–external fins for a phase change material (PCM) in duplex and triplex tube heat exchangers (DTHX and TTHX) are investigated numerically. A two-dimensional analysis is carried out using ANSYS Fluent for various configurations. Moreover, the effect of different critical parameters, number of fins, fin length, fin thickness, and the heat exchanger tube material are evaluated in terms of the total time of complete phase change of the PCM. Four cases are investigated; cases 1 and 2 are based upon a DTHX while cases 3 and 4 are TTHXs. By considering case 1 as a reference case, it is found that case 2 and case 3 reduce the total melting time by 48.76% and 90.12%, respectively. Case 4 achieves the shortest time for complete melting of the PCM, and the total melting time is decreased by 92%. Solidification behaviour for all four cases is also investigated. The novel configurations increase (doubled) the supply of heat transfer fluid (HTF) while at the same time significantly enhance the melting/solidification characteristics for all the cases without disrupting the convectional currents during phase change of the PCM. Tube materials with different thermophysical properties are also investigated with the heat transfer rate and melting time significantly improved with a high thermal diffusivity material. Moreover, the heat transfer is found to increase with fin length and fin thickness.