  Given a surface $\Sigma$ in $\mathbb{R}^3$ diffeomorphic to $S^2$, Struwe
(Acta Math., 1988) proved that for almost every $H$ below the mean curvature of
the smallest sphere enclosing $\Sigma$, there exists a branched immersed disk
which has constant mean curvature $H$ and boundary meeting $\Sigma$
orthogonally. We reproduce this result using a different approach and improve
it under additional convexity assumptions on $\Sigma$. Specifically, when
$\Sigma$ itself is convex and has mean curvature bounded below by $H_0$, we
obtain existence for all $H \in (0, H_0)$. Instead of the heat flow used by
Struwe, we use a Sacks-Uhlenbeck type perturbation. As in previous joint work
with Zhou (arXiv:2012.13379), a key ingredient for extending existence across
the measure zero set of $H$'s is a Morse index upper bound.
