A novel method, Hamiltonian scaling grand canonical Monte Carlo, has been used to determine the phase behavior of 12 variations of the modified Buckingham exponential-6 potential. Hamiltonian scaling grand canonical Monte Carlo enables the determination of thermodynamic properties of several related potential models from a single set of simulations. The main advantage of the method is that by appropriate selection of the chemical potential and temperature for each Hamiltonian, the densities that are sampled are relevant for multiple Hamiltonians, thus increasing the simulation efficiency. The method was combined with mixed-field finite-size scaling to determine the critical parameters and phase coexistence curves of several modified Buckingham exponential-6 potentials to a high level of accuracy. In particular, the critical parameters of the models have been calculated to within 0.3% for the critical temperature and 1% for the critical density and pressure. A new potential;for methane based on the Buckingham exponential-6 model is presented. The vapor pressure curve of methane from the Buckingham exponential-6 potential model is much more accurate than is possible with the Lennard-Jones potential model.