Let $\Gamma$ be a smooth Jordan arc, and let $\Phi \colon \hat{\mathbb C} \backslash \overline{\mathbb D} \to \hat{\mathbb C} \backslash \Gamma$ be a conformal isomorphism that fixes the point at $\infty$. Then $\Phi$ extends continuously onto $\partial \mathbb D$ because $\Gamma$ is smooth (Caratheodory), and for every $z \in \Gamma$ except the two endpoints, $\Phi^{-1}(z)$ consists of two points on $S^1$.

Now if I have another smooth Jordan arc $\Gamma_0$, and a conformal isomorphism $\Phi_0 \colon \hat{\mathbb C} \backslash \overline{\mathbb D} \to \hat{\mathbb C} \backslash \Gamma_0$ that fixes $\infty$ such that $\Phi_0(\Phi^{-1}(z))$ is a singleton for every $z \in \Gamma$, can I necessarily conclude that $\Gamma$ and $\Gamma_0$ are the same curve (modulo affine transformation)?