Summary: A connected graph \(G=(V,E)\) is called a quasi-tree (resp. a quasi-unicyclic graph), if there exists \(u_0\in V(G)\) such that \(G-u_0\) is a tree (resp. a unicyclic graph). Let \(\lambda_1(G),\lambda_2(G),\dots ,\lambda_n(G)\) be the eigenvalues of the adjacency matrix \(A(G)\) of \(G\) and \(S(G)=(S_0(G),S_1(G),\dots ,S_{n-1}(G))\) the sequence of spectral moments of a graph \(G\), where \(S_k(G)=\sum_{i=1}^n\lambda_i^k(G)\) \((k=0,1,\dots ,n-1)\) is the \(k\)-th spectral moment of \(G\). For two graphs \(G_1\), \(G_2\), we have \(G_1\prec_SG_2\), if for some \(k\) \((k=1,2,\dots ,n-1)\), we have \(S_i(G_1)=S_i(G_2)\) \((i=0,1,\dots ,k-1)\) and \(S_k(G_1)<S_k(G_2)\). In [Linear Algebra Appl. 436, No. 5, 927–934 (2012; Zbl 1239.05117)], F. X. Pan et al. first determined the last and the second last quasi-tree, in an \(S\)-order, in the set of all quasi-tree with \(n\) vertices. Motivated by this paper, we focus on this problem, and present a new approach, with a shorter proof, to determine the last graphs in the \(S\)-order in all the quasi-trees. Furthermore, we identify the last graph in the set of quasi-unicyclic graphs, in an \(S\)-order, by this approach.