The diminished Sombor index of a graph \( G \) with edge set \( E(G) \) is defined as \[ DSO(G) = \sum_{uv \in E(G)} \frac{\sqrt{d_u^2 + d_v^2}}{d_u + d_v}, \] where \( d_u \) denotes the degree of vertex \( u \). In this paper, we revisit and refine some of the results reported in the recent paper [MATCH Commun. Math. Comput. Chem. 95 (2026) 141--162]. One of the obtained refined results guarantees that \( DSO(G) \) decreases when any of the edges of \( G \) is removed. Also, one of the new results gives the graphs minimizing \( DSO \) over the class \( \mathcal{G}_{m,n} \) of all connected graphs of order \( n \) and size \( m \) for \( 3n\ge 2m\ge 2(n+2) \). The paper is concluded with an open problem concerning the graphs minimizing \( DSO \) over \( \mathcal{G}_{m,n} \) for \( m\ge\max\left\{n+3,\lceil 3n/2\rceil\right\} \).