Let F be a non-Archimedean local field. For any irreducible smooth representation \(\pi \) of \(\textrm{GL}_n(F)\) and a multisegment \({\mathfrak {m}}\) , we have an operation \(D_{{\mathfrak {m}}}(\pi )\) to construct a simple quotient \(\tau \) of a Bernstein-Zelevinsky derivative of \(\pi \) . This article continues the previous one to study the following poset \(\begin{aligned} {\mathcal {S}}(\pi , \tau ) {:}{=}\left\{ {\mathfrak {n}} : D_{{\mathfrak {n}}}(\pi )\cong \tau \right\} , \end{aligned}\) where \({\mathfrak {n}}\) runs for all the multisegments. Here the partial ordering on \({\mathcal {S}}(\pi , \tau )\) comes from the Zelevinsky ordering. We show that the poset has a unique minimal multisegment. Along the way, we introduce two new ingredients: fine chain orderings and local minimizability.
