  The Hall effect is recently shown to be efficient in magnetized dense
molecular cores, and could lead to a bimodal formation of rotationally
supported discs (RSDs) in the first core phase. However, how such Hall
dominated systems evolve in the protostellar accretion phase remains unclear.
We carry out 2D axisymmetric simulations including Hall effect and Ohmic
dissipation, with realistic magnetic diffusivities computed from our
equilibrium chemical network. We find that Hall effect only becomes efficient
when the large population of very small grains (VSGs: $\lesssim$10 nm) is
removed from the standard MRN size distribution. With such an enhanced Hall
effect, however, the bimodality of disc formation does not continue into the
main accretion phase. The outer part of the initial $\sim$40 AU disc formed in
the anti-aligned configuration (${\bf \Omega \cdot B}<0$) flattens into a thin
rotationally supported Hall current sheet as Hall effect moves the poloidal
magnetic field radially inward relative to matter, leaving only the inner
$\lesssim$10--20 AU RSD. In the aligned configuration (${\bf \Omega \cdot
B}>0$), disc formation is suppressed initially but a counter-rotating disc
forms subsequently due to efficient azimuthal Hall drift. The counter-rotating
disc first grows to $\sim$30 AU as Hall effect moves the magnetic field
radially outward, but only the inner $\lesssim$10 AU RSD is long-lived like in
the anti-aligned case. Besides removing VSGs, cosmic ray ionization rate should
be below a few 10$^{-16}$ s$^{-1}$ for Hall effect to be efficient in disc
formation. We conclude that Hall effect produces small $\lesssim$10--20 AU
discs regardless of the polarity of the magnetic field, and that radially
outward diffusion of magnetic fields remains crucial for disc formation and
growth.
