This study deals with the unsteady close-contact melting of solid blocks on a flat surface subject to convective heating. Normalizing the model equations in reference to the steady solution successfully leads them to cover constant heat flux and isothermal limits at small and large extremes of the Biot number, respectively. The resulting equations admit a compactly expressed analytical solution, which includes the previous solutions as a subset. Based on the steady solution, the characteristics of close-contact melting can be categorized into constant heat flux, transition, and isothermal regimes, the boundaries of which appear to be nearly independent of the contact force. The unsteady solutions corresponding to Biot numbers in the transition regime show intermediate behaviors between those of the two limits. With a proper approximation, the present solution procedure can cope with the case of variable fluid temperature and heat transfer coefficient. Regardless of imposed conditions, the mean normalized Nusselt number during the unsteady process asymptotically approaches to a constant value as the Biot number comes close to each limit.