  We explore different gauge choices in the moving puncture formulation in
order to improve the accuracy of a linear momentum measure evaluated on the
horizon of the remnant black hole produced by the merger of a binary. In
particular, motivated by constant values studies, we design a gauge via a
variable shift parameter $m\eta(\vec{r}(t))$ such that it takes a low
asymptotic (and at the orbiting punctures) value, while about the standard
value of 2 at the final hole horizon. This choice then follows the remnant
black hole as it moves due to its net recoil velocity. We find that this choice
keeps the accuracy of the binary evolution and, once the asymptotic value of
the parameter $m\eta$ is chosen about or below 1.0, it produces more accurate
results for the recoil velocity than the corresponding evaluation of the
radiated linear momentum at infinity, for typical numerical resolutions. We
also find that the choice of the $\partial_t$-gauge (at our working
resolutions) is more accurate in this regard of computing recoil velocities
than the $\partial_0$-gauge. Detailed studies of an unequal mass
$q=m_1/m_2=1/3$ nonspinning binary are provided and then verified for other
mass ratios $(q=1/2,1/5)$ and spinning $(q=1)$ binary black hole mergers.
