Converting a $1$-connected simplicial complex into a $1$-reduced simplicial set

There is a natural way to convert a simplicial complex $C$ into an "equivalent" simplicial set $S$: after ordering the vertices, the simplices in $C$ correspond exactly to the non-degenerate simplices of $S$.

If $C$ is finite and connected, it can easily be converted to a $0$-reduced sset $S$ (that is, having only one vertex) by factoring out a maximal tree in the $1$-skeleton: the underlying topological spaces have the same homotopy type.

Can a similar method be applied to convert a simply connected complex $C$ into a $1$-reduced sset (that is, having only one vertex and only one $1$-simplex)? I guess that for this, we need to have an algorithm for contracting loops in $C$. Let's say that for each loop $\gamma$, a simplicial map $f_{\gamma}$ is given from a subdivision $D$ of a $2$-disc into $C$ such that $f|_{\partial D}=\gamma$ (or something like that).

Is there an algorithmic way how to use the loop-contractions to convert $S$ into something $1$-reduced and of the same homotopy type?

The intuition is simple: "factor out the filler of each loop". But I have problem formalizing it. It seems to me that the filler can be complicated, self-intersecting, etc, not anything that simple as a spanning tree.