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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$σ$. 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.