Classical many-body potentials that have a rather low number of parameters and are based on physically-inspired functional forms are expected to display a reasonable transferability, though their flexibility is limited. Ways to improve them when their transferability is unsatisfactory and when new interesting structures/targeted properties emerge are not clearly established. Here, model screening and sensitivity analysis techniques are combined to get insights on the intrinsic capabilities of classical potentials for typical sets of targeted properties, and we propose to use the outcomes of the sensitivity analysis to improve potential performances. Usefulness is illustrated on existing simple second moment potentials (SMA), and we refit one such model for the study of small irradiation defects in α-zirconium. Application of this approach is recommended for more complex interaction models with up to tens of parameters, such as other classical many-body potentials or tight-binding electronic structure models, that are possible to re-optimize while gaining a better understanding of the role of their parameters.