  JWST observations demonstrate that supermassive black holes (SMBHs) exist by
redshifts $z \gtrsim 10$, providing further evidence for "direct collapse"
black hole (BH) formation, whereby massive ($\sim 10^{3-5} M_{\odot}$) SMBH
seeds are generated within a few Myr as a byproduct of the rapid inflow of gas
into the centers of protogalaxies. Here we analyze the intermediate
"quasi-star" phase that accompanies some direct collapse models, during which a
natal BH accretes mass from and energetically sustains (through accretion) an
overlying gaseous envelope. We argue that previous estimates of the maximum BH
mass that can be reached during this stage, $\sim 1\%$ of the total quasi-star
mass, are unphysical, and arise from underestimating the efficiency with which
energy can be transported outward from regions close to the BH. We construct
new quasi-star models that consist of an inner, "saturated-convection" region
(which conforms to a convection-dominated accretion flow near the BH) matched
to an outer, adiabatic envelope. These solutions exist up to a BH mass of $\sim
60\%$ the total quasi-star mass, at which point the adiabatic envelope contains
only 2\% of the mass (with the remaining $\sim 38\%$ in the
saturated-convection region), and this upper limit is reached within a time of
$20-40$ Myr. We conclude that quasi-stars remain a viable route for producing
SMBHs at large redshifts, consistent with recent JWST observations.
