There is a well known theorem which states that:
Theorem(Stallings): For any immersion $f$ from a finite graph $D$ to $G$ there is a finite-sheeted covering space $D '$ of $G$ that extends $f$. More precisely, there is an embedding of $D$ into $D'$ and the restriction of the covering map to $D$ coincides with $f$.
This implies that every f.g. subgroup $H$ of a free group $F$, then $H$ is a free factor in a finite-index subgroup of $F$.
My question is if we have a group that is a free product of groups, which has finite Kurosh rank (it can be written as a free product of finitely many freely indecomposable groups), then if there is in the literature something similar to the Stallings' theorem, using graph of groups or Bass- Serre trees instead of finite graphs.
To be more precise, if we start with a graph of groups $\Gamma$ which corresponds to a free product decomposition of $G$, so in particular it has trivial edges stabilisers, then is there something like 'finite covering' of $\Gamma$ which corresponds to finite index subgroups of $G$, and starting with any immersion of a graph of groups to $\Gamma$ is it possible to be completed to a 'finite cover'?
If anyone knows something similar or at least some special cases, it would be very helpful. Thanks a lot in advance.