Let $M$ be a smooth surface, let $r \geq 1$, and let $f: M \to M$ be a $C^r$ diffeomorphism. Let $p \in M$ be a hyperbolic fixed point of saddle type, with one-dimensional local stable and unstable manifolds $W^s_{\mathrm{loc}}(p)$ and $W^u_{\mathrm{loc}}(p)$ chosen inside a coordinate neighbourhood of $p$. Suppose $\gamma \subset M$ is the image of an embedded $C^r$ arc and that $\gamma$ intersects $W^s_{\mathrm{loc}}(p)$ transversely at a point $x \in \gamma \cap W^s_{\mathrm{loc}}(p)$. Assume the local stable manifold is chosen so that $f^m(x) \to p$ and, for every sufficiently small subarc of $\gamma$ through $x$, some forward iterate enters the local chart as an admissible $C^1$ graph transverse to the stable cone. Assume further that in this chart the standard local graph transform for $f$ is defined on admissible $C^1$ graphs, contracts their height and slope toward the unstable axis, and expands their unstable-coordinate domains so that the domains eventually contain every prescribed compact subinterval of the local unstable axis.

Then for every compact subarc $J \subset W^u_{\mathrm{loc}}(p)$ and every $C^1$ neighbourhood $\mathcal U$ of $J$ in the space of embedded $C^1$ arcs, there exists $N \in \mathbb N$ such that for every $n \geq N$, the curve $f^n(\gamma)$ contains a $C^1$ subarc $\gamma_n$ with $\gamma_n \in \mathcal U$.