A modification of an embedded-atom method (EAM)-type potential is proposed for a quantitative description of equilibrium and nonequilibrium properties of metal systems within the molecular dynamics framework. The modification generalizes the previously developed linear correction to EAM-type potentials [Sushko et al., J. Phys.: Condens. Matter 28 (2016) 145201] and asymptotically approaches zero at large interatomic distances. A general procedure for constructing this modification is outlined and its relation to the linear correction is elaborated. To benchmark this procedure, we examine the melting phase transition and several equilibrium properties of finite-size nanosystems made of silver, gold and titanium. The simulations performed with the modified potential predict higher bulk melting temperatures of the metals and agree better with experimental values as compared to the original EAM-type potential. Our results show that the modification works well for metals with both cubic and hexagonal lattice structures. The Gupta potential is chosen as an example but the modification proposed can also be applied to other potentials of the EAM type.