  The joint observation of the gravitational-wave and electromagnetic signal
from the binary neutron-star merger GW170817 allowed for a new independent
measurement of the Hubble constant $H_0$, albeit with an uncertainty of about
15\% at 1$\sigma$. Observations of similar sources with a network of future
detectors will allow for more precise measurements of $H_0$. These, however,
are currently largely limited by the intrinsic degeneracy between the
luminosity distance and the inclination of the source in the gravitational-wave
signal. We show that the higher-order modes in gravitational waves can be used
to break this degeneracy in astrophysical parameter estimation in both the
inspiral and post-merger phases of a neutron star merger. We show that for
systems at distances similar to GW170817, this method enables percent-level
measurements of $H_0$ with a single detection. This would permit the study of
time variations and spatial anisotropies of $H_0$ with unprecedented precision.
We investigate how different network configurations affect measurements of
$H_0$, and discuss the implications in terms of science drivers for the
proposed 2.5- and third-generation gravitational-wave detectors. Finally, we
show that the precision of $H_0$ measured with these future observatories will
be solely limited by redshift measurements of electromagnetic counterparts.
