For a graph $G$, the general reduced second Zagreb index is defined as $$GRM_λ(G) = \sum_{uv \in E} (deg(u) + λ) (deg(v) + λ),$$ where $λ$ is an arbitrary real number and $deg (v)$ is the degree of the vertex $v$. In this paper, we extend and correct the equality results from [N. Dehgardia, S. Klav\v zar, {\it Improved lower bounds on the general reduced second Zagreb index of trees}, preprint (2023)] regarding the minimal value of $GRM_λ$ for $λ\geq -1$ among trees with $n$ vertices and a maximal degree $Δ$. Furthermore, we complement these results with two distinct approaches to determine the minimum value of the general reduced second Zagreb index for molecular trees with $Δ= 3$ and $Δ= 4$ in $λ= -2$, and characterize the extremal trees.