  Super-massive black holes weighing up to $\sim 10^9 \, \mathrm{M_{\odot}}$
are in place by $z \sim 7$, when the age of the Universe is $\lesssim 1 \,
\mathrm{Gyr}$. This implies a time crunch for their growth, since such high
masses cannot be easily reached in standard accretion scenarios. Here, we
explore the physical conditions that would lead to optimal growth wherein
stable super-Eddington accretion would be permitted. Our analysis suggests that
the preponderance of optimal conditions depends on two key parameters: the
black hole mass and the host galaxy central gas density. In the high-efficiency
region of this parameter space, a continuous stream of gas can accrete onto the
black hole from large to small spatial scales, assuming a global isothermal
profile for the host galaxy. Using analytical initial mass functions for black
hole seeds, we find an enhanced probability of high-efficiency growth for seeds
with initial masses $\gtrsim 10^4 \, \mathrm{M_{\odot}}$. Our picture suggests
that a large population of high-$z$ lower-mass black holes that formed in the
low-efficiency region, with low duty cycles and accretion rates, might remain
undetectable as quasars, since we predict their bolometric luminosities to be
$\lesssim 10^{41} \, \mathrm{erg \, s^{-1}}$. The presence of these sources
might be revealed only via gravitational wave detections of their mergers.
