In a recent paper titled Inverse matroid optimization under subset constraints, we consider generalizations of the classical Inverse Matroid problem (IM). In IM, we are given a matroid, a fixed basis $B$, and an initial weight function, and the goal is to minimally modify the weights — measured by some function — so that $B$ becomes a maximum-weight basis. The problem arises naturally in settings where one wishes to explain or enforce a given solution by minimally perturbing the input. We extend this classical problem by replacing the fixed basis with a subset $S_0$ of the ground set and imposing various structural constraints on the set of maximum-weight bases relative to $S_0$. Specifically, we study six variants: (A) Inverse Matroid Exists, where $S_0$ must contain at least one maximum-weight basis; (B) Inverse Matroid All, where all bases contained in $S_0$ are maximum-weight; and (C) Inverse Matroid Only, where $S_0$ contains exactly the maximum-weight bases, along with their natural negated counterparts. For all variants, we develop combinatorial polynomial-time algorithms under the $\ell_\infty$-norm. A key ingredient is a refined min-max theorem for Inverse Matroid under the $\ell_\infty$-norm, which enables simpler and faster algorithms than previous approaches and may be of independent combinatorial interest.