The recently introduced [Mol. Phys. 105, 663 (2007)] 'Morse/long-range' (or MLR) potential energy function is a very flexible form which explicitly incorporates the theoretically predicted inverse-power-sum long-range tail, is smooth and differentiable everywhere, and includes the well depth ? , equilibrium distance r e, and long-range interaction coefficients Cm as explicit parameters. This form is being used increasingly commonly in direct-potential-fit analyses of experimental data. The present work shows that the MLR form can readily accommodate the inclusion of 'damping functions' in the description of the long-range potential tail, and that inclusion of such terms leads to much more realistic short-range extrapolation behaviour. Illustrative applications to the ground electronic states of MgH, Li2 and ArXe are presented.