We study the "almost-diameter" of a hyperbolic surface $X$, i.e. the smallest radius of a hyperbolic ball around a point $x_0$ which covers almost all of $X$. This question is closely related to the works of Parzanchevski-Sarnak on Golden Gates, and Gosh-Gorodnik-Nevo on Diophantine Exponents. We show that it follows from Selberg's conjecture that the almost-diameter of the surfaces corresponding to the principal congruence subgroups of $\textrm{SL}_2(\mathbb{Z})$ have optimal almost-diameter. Following the work of Sarnak-Xue, we also prove this unconditionally, and explain how our method can be generalized to other similar problems. Joint work with Konstantin Golubev.