Let \( G \) be a connected graph having more than two vertices and let \( d_i \) denote the degree of vertex \( v_i \) in \( G \). Let \( E(G) \) represent the edge set of \( G \). Then, the augmented Sombor (ASO) index of \( G \) is defined as \( ASO(G) = \sum_{v_i v_j \in E(G)} \sqrt{(d_i + d_j - 2)^{-1}(d_i^2 + d_j^2)}. \) It is known that the cycle graph \( C_n \) uniquely minimizes the ASO index in the class of all \( n \)-order unicyclic graphs. In this paper, we prove that the unique \( n \)-order unicyclic graph of maximum degree \( n-1 \) maximizes the ASO index in the aforementioned unicyclic graph class. We also prove that \( ASO(G-v_iv_j) < ASO(G) \) whenever neither of the graphs \( G-v_iv_j \) and \( G \) contains any isolated edge. Utilizing this edge-deletion property, we characterize the unique graph maximizing the ASO index among all fixed-order connected graphs with a specified vertex connectivity (or edge connectivity).