Let \( G \) be a graph with vertex set \( V \) and edge set \( E \). A topological index has the form \[ TI=TI(G)=\sum_{uv\in E}f(d_G(u),d_G(v)), \] where \( f=f(x,y) \) is a pertinently chosen function which must be symmetric and real-valued for all \( x,y \) pertaining to vertex degrees of the graph \( G \). Particularly interesting are the Sombor index and the elliptic Sombor index, defined by the functions \( f(x,y)=\sqrt{x^2+y^2} \) and \( f(x,y)=(x+y)\sqrt{x^2+y^2} \), respectively. Let \( q=2f(2,3)-f(2,2)-f(3,3) \). In this paper, we characterize the extremal graphs that achieve the upper bounds of the topological index \( TI \) for benzenoid systems, where \( TI \) satisfies the conditions \( 0 < q < \frac{f(2,2)}{2} \) or \( -\frac{f(2,2)}{4} < q < 0 \), respectively. In addition, we provide a lower bound for the Sombor index on benzenoid systems.