  In a previous work, we have identified the spin of the dominant black hole of
a binary from its jet properties. Analysing Very Long Baseline Array (VLBA)
observations of the quasar S5~1928+738, taken at $15$-GHz during $43$ epochs
between $1995.96$ and $2013.06$, we showed that the inclination angle variation
of the inner ($<2$~mas) jet symmetry axis naturally decomposes into a periodic
and a monotonic contribution. The former emerges due to the Keplerian orbital
evolution, while the latter is interpreted as the signature of the spin-orbit
precession of the jet emitting black hole. In this paper, we revisit the
analysis of the quasar S5~1928+738 by including new $15$-GHz VLBA observations
extending over $29$ additional epochs, between $2013.34$ and $2020.89$. The
extended data set confirms our previous findings which are further supported by
the flux density variation of the jet. By applying an enhanced jet precession
model that can handle arbitrary spin orientations $\kappa$ with respect to the
orbital angular momentum of a binary supermassive black hole system, we
estimate the binary mass ratio as $\nu=0.21\pm0.04$ for $\kappa=0$ (i.e. when
the spin direction is perpendicular to the orbital plane) and as
$\nu=0.32\pm0.07$ for $\kappa=\pi/2$ (i.e. when the spin lies in the orbital
plane). We estimate more precisely the spin precession velocity, halving its
uncertainty from $(-0.05\pm0.02)^\circ\,\mathrm{yr}^{-1}$ to
$(-0.04\pm0.01)^\circ\,\mathrm{yr}^{-1}$.
