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Binary neutron star mergers offer a new and independent means of measuring the Hubble constant $H_0$ by combining the gravitational-wave inferred source luminosity distance with its redshift obtained from electromagnetic follow-up. This method is limited by intrinsic degeneracy between the system distance and orbital inclination in the gravitational-wave signal. Observing the afterglow counterpart to a merger can further constrain the inclination angle, allowing this degeneracy to be partially lifted and improving the measurement of $H_0$. In the case of the binary neutron star merger GW170817, afterglow light-curve and imagery modeling thus allowed to improve the $H_0$ measurement by a factor of 3. However, systematic access to afterglow data is far from guaranteed. In fact, though each one allows a leap in $H_0$ precision, these afterglow counterparts should prove rare in forthcoming multimessenger campaigns. We combine models for emission and detection of gravitational-wave and electromagnetic radiation from binary neutron star mergers with realistic population models and estimates for afterglow inclination angle constraints. Using these models, we quantify how fast $H_0$ will be narrowed-down by successive multimessenger events with and without the afterglow. We find that, because of its rareness and though it greatly refines angle estimates, the afterglow counterpart should not significantly contribute to the measurement of $H_0$ in the long run.