  We compute hybrid static potentials in $SU(3)$ lattice Yang-Mills theory at
short quark-antiquark separations using four different small lattice spacings
as small as $0.04\,\text{fm}$. The resulting static potentials are important,
e.g. when studying heavy hybrid mesons in the Born-Oppenheimer approximation.
We also discuss and exclude possible systematic errors from topological
freezing, the finite lattice volume and glueball decays.
