How post order relates to strongly connected components
Let $G = (V,E)$ be a directed graph with two strongly connected components $S$ and $S'$. Let $post: V \rightarrow \mathbb{N}$ be the result of running the DFS to find connected components in an undirected graph algorithm $A$. If there exists $v \in S$ and $w \in S'$ such that $(v,w) \in E$ then
$$\max_{x \in S} post(x) > \max_{y \in S'} post(y).$$Proof
As $S$ and $S'$ are separate strongly connected components with an edge $(S, S')$ in the strongly connected component graph. As the strongly connected component graph is a DAG there is not a path from $S'$ to $S$ in $G$.
During the fun of DFS to find connected components in an undirected graph $A$ we must visit either a vertex in $S$ or $S'$ first.
Suppose we first visit a vertex $w \in S'$. Here all $S'$ will be visited by explore from $w$ and the algorithm will have to leave via $w$ before it can visit a vertex of $S$. Therefore all vertices in $S'$ will have lower post number than any vertex in $S$.
Suppose instead we first visit a vertex $w \in S$. From this vertex we will explore the whole of $S$ and $S'$ before exiting $w$. Therefore all vertices in $S$ and $S'$ will appear in a branch below $w$ in the DFS tree. This gives that $w$’s post number is larger than that for all vertices $S \cup S' \backslash \{w\}$. So
$$post(w) = \max_{x \in S} post(x) > \max_{y \in S'} post(y)$$as required.