Direct reduction from $st\text{-}non\text{-}connectivity$ to $st\text{-}connectivity$
It is possible, if messy, to convert the proof of the Immerman-Szelepcsényi theorem to the reduction you want. There is absolutely no need to use the NL-completeness of st-connectivity.

Given an instance $G=(V,E),s,t$, we construct a new graph $G'=(V',E'),s',t'$. The "major vertices" of $V'$ record the following information: the current distance $d$ from $s$, the number of vertices of distance at most $d-1$, the number of vertices of distance $d-1$ counted so far, the current vertex we're guessing if it has distance at most $d-1$, the number of vertices of distance at most $d$ counted so far, the current vertex we're determining whether it has distance at most $d$. The minor vertices handle the part where we guess a path of length at most $d-1$ to a vertex which we guess to be of distance at most $d-1$. Edges that involve showing the vertex $t$ is reachable from $s$ are dropped. For each vertex which we're testing at the current distance, we only move forward to the next vertex if we have accounted for all vertices of smaller distance. When moving from distance $d$ to distance $d+1$, we copy the requisite information. The starting vertex $s'$ accounts for the fact that $s$ is the only vertex of distance zero. The ending vertex $t'$ is pointed at by all vertices representing the fact that the process has finished up to (and including) distance $n-1$, where $n=|V|$.

As you can see, it will be quite messy to write everything in full and correctly, but definitely possible. No overt use of NL-completeness was made, in that we never use the configuration graph of any NL machine. That's not needed, since we have something better than the configuration graph - the input instance itself.