We study the traveling salesman problem in the hyperbolic plane of Gaussian curvature $-1$. Let $α$ denote the minimum distance between any two input points. Using a new separator theorem and a new rerouting argument, we give an $n^{O(\log^2 n)\max(1,1/α)}$ algorithm for Hyperbolic TSP. This is quasi-polynomial time if $α$ is at least some absolute constant, and it grows to $n^{O(\sqrt{n})}$ as $α$ decreases to $\log^2 n/\sqrt{n}$. (For even smaller values of $α$, we can use a planarity-based algorithm of Hwang et al. (1993), which gives a running time of $n^{O(\sqrt{n})}$.)