  In 2013, Zhai proved that most numerical semigroups of a given genus have
depth at most $3$ and that the number $n_g$ of numerical semigroups of a genus
$g$ is asymptotic to $S\varphi^g$, where $S$ is some positive constant and
$\varphi \approx 1.61803$ is the golden ratio. In this paper, we prove
exponential upper and lower bounds on the factors that cause $n_g$ to deviate
from a perfect exponential, including the number of semigroups with depth at
least $4$. Among other applications, these results imply the sharpest known
asymptotic bounds on $n_g$ and shed light on a conjecture by Bras-Amor\'os
(2008) that $n_g \geq n_{g-1} + n_{g-2}$. Our main tools are the use of Kunz
coordinates, introduced by Kunz (1987), and a result by Zhao (2011) bounding
weighted graph homomorphisms.
