Log-Space Reduction $USTCON\le_L CO-2Col$
Let USTCON2 be the following problem: given a graph $G$ and two vertices $s,t$, decide whether there is a walk from $s$ to $t$ of even length. Let us show first that USTCON2 reduces to co2COL.

Given a graph $G = (V,E)$ and two vertices $s,t$, construct a new graph whose vertex set consists of two copies of $V$. For every $(a,b) \in E$, connect $a$ in the first copy to $b$ in the second copy, and $a$ in the second copy to $b$ in the first copy. So far the graph is bipartite. Now connect $s$ and $t$ in the first copy. The new graph is bipartite iff there is no path of even length from $s$ to $t$ in the original graph.

Now let us reduce USTCON to USTCON2. Given a graph $G = (V,E)$ and two vertices $s,t$, add two new vertices $x,y$, and edges $(s,x),(x,y),(y,s)$. There is a walk from $s$ to $t$ in the original graph iff there is a walk of even length from $s$ to $t$ in the new graph.