Based on the above comment of Holger Partsch, you can't factor a lot of stuff.  You can do silly things like:

A necessary condition for factoring $f$ through a given $S'$ is that you can embed:
$$K(S) \subseteq K(S') \subseteq K(X).$$  

If you are willing to blow-up $X$ to resolve indeterminacies, then this might be the only obstruction in the geometric setting anyway (assuming $S$ and $S'$ are normal).  

On the other hand [THIS][1] question, and the numerous excellent answers, seems like it might be very relevant depending on your context.


  [1]: https://mathoverflow.net/questions/35429/which-algebraic-varieties-admit-a-morphism-to-a-curve