We show that any matrix perturbation of an $n \times n$ nilpotent complex matrix is similar to a matrix perturbation whose leading coefficient has minimal Jordan structure. Additionally, we derive the property that, for matrix perturbations with minimal leading Jordan structure, the sufficient conditions of Lidskii's perturbation theorem for eigenvalues are necessary too. It is further shown how minimality can be obtained by computing a similarity transform whose entries are polynomials of degree at most $n$. This relies on an extension of both Lidskii's theorem and its Newton diagram-based interpretation.