A remark on $\epsilon$-maps in dimension 1

Let $f\text{\hspace{0.17em}⁣}:{\mathbb{S}}^1\to G$ be a surjective map from the standard unit circle to a graph $G$ such that the pre-image of each point has diameter less than $\epsilon$. If $\epsilon$ is small enough, does $f$ split as a free factor in ${\pi }_1(G)$?