Numerical modeling of efficient close contact melting (CCM) has been a significant challenge in phase-change heat transfer research. This work addresses this issue by developing a modified equivalent heat capacity method to construct a universal CCM model, considering physicals such as conduction, heat convection, and solid sinking. The enhanced heat transfer mechanism of CCM compared to conventional constrained melting (CM) in finned latent heat storage (LHS) units is investigated. The results reveal that the early stages of CM and CCM are dominated by heat conduction, while natural convection and mixed convection are induced in the late stage of CM and CCM, respectively. The CM and CCM performance is optimized under upward and downward flows, respectively. The complete melting time of CCM is reduced by a substantial 36.7% and 29.7% compared to CM for both downward and upward flow cases. The inlet temperature and volume flow rate are positively correlated with CM and CCM performance, but the effect of volume flow rates is smaller. The gradient fin maximizes melting performance at a height gradient of 1. Compared to the uniform layout, the complete melting time of CM and CCM in the optimized gradient LHS unit is reduced by 5.6% and 8.5%.