Phase change materials (PCMs) can absorb large amounts of heat without significant rise of their temperature during the melting process. This effect is attractive for using in heat storage, e.g. for solar-thermal power plants. Heat transfer in PCMs, which have low thermal conductivity, can be enhanced by fins that enlarge the heat transfer area. However, when the PCM melts, a layer of liquid is growing at the fins creating an increasing thermal resistance that impedes the process. The present work demonstrates that performance of a latent-heat thermal storage unit may be considerably affected by achieving a so-called close-contact melting (CCM) which occurs when the solid phase is approaching a heated surface, and only a thin liquid layer is separating between the two. Although CCM was extensively studied in the past, its possible role in finned systems has been revealed only recently by our group. In particular, it depends heavily on the specific configuration of the fins. In the present work, some new findings in the field are reported. It is demonstrated that close-contact melting shortens the melting time drastically, provided a proper orientation of the fins is chosen. The results may be expressed in terms of the time-dependent melt fraction and Nusselt number, showing their dependence on the Stefan and Fourier numbers. Moreover, the approach used in the present study may be applied to geometries in which the heated surface is not horizontal or where there are a number of heated surfaces or fins.