  We construct new equations of state for baryons at sub-nuclear densities for
the use in core-collapse supernova simulations. The abundance of various nuclei
is obtained together with thermodynamic quantities. The formulation is an
extension of the previous model, in which we adopted the relativistic mean
field theory with the TM1 parameter set for nucleons, the quantum approach for
$d$, $t$, $h$ and $\alpha$ as well as the liquid drop model for the other
nuclei under the nuclear statistical equilibrium. We reformulate the model of
the light nuclei other than $d$, $t$, $h$ and $\alpha$ based on the
quasi-particle description. Furthermore, we modify the model so that the
temperature dependences of surface and shell energies of heavy nuclei could be
taken into account. The pasta phases for heavy nuclei and the Pauli- and
self-energy shifts for $d$, $t$, $h$ and $\alpha$ are taken into account in the
same way as in the previous model. We find that nuclear composition is
considerably affected by the modifications in this work, whereas
thermodynamical quantities are not changed much. In particular, the washout of
shell effect has a great impact on the mass distribution above $T \sim 1$ MeV.
This improvement may have an important effect on the rates of electron captures
and coherent neutrino scatterings on nuclei in supernova cores.
