  Let $X$ be a smooth Fano variety over $\mathbb{C}$ and let $B$ be a smooth
projective curve over $\mathbb{C}$. Geometric Manin's Conjecture predicts the
structure of the irreducible components $M \subset \mathrm{Mor}(B, X)$
parametrizing curves which are non-free and have large anticanonical degree.
Following ideas of our previous work, we prove the first prediction of
Geometric Manin's Conjecture describing such irreducible components. As an
application, we prove that there is a proper closed subset $V \subset X$ such
that all non-dominant components of $\mathrm{Mor}(B, X)$ parametrize curves in
$V$, verifying an expectation put forward by Victor Batyrev. We also
demonstrate two important ways that studying $\mathrm{Mor}(B,X)$ differs from
studying the space of sections of a Fano fibration $\mathcal{X} \to B$.
