  In this paper, the implications of the symmetry energy on the hadron and
quark phase transitions in the compact star, including the properties of the
possible configurations of the quark-hadron hybrid stars, are investigated in
the frameworks of the energy-density functional (EDF) models and the flavor
SU(2) Nambu--Jona-Lasinio (NJL) model with the help of the Schwinger's
covariant proper-time regularization (PTR) scheme. In this {theoretical setup},
the equations of states (EoSs) of hadronic matter for various values of
symmetry energies obtained from the EDF models are employed to describe the
hadronic matter, and the {flavor} SU(2) NJL model with various repulsive-vector
interaction strengths are used to describe the quark matter. We then observe
the obtained EoS in the mass-radius properties of the hybrid star
configurations for various vector interactions and nuclear symmetry energies by
solving the Tolman-Oppenheimer-Volkoff equation. We obtain that the critical
density at which the phase transition occurs varies over the density
(3.6--6.7)$\rho_0$ depending on the symmetry energy and the strength of the
vector coupling $G_v$. The maximum mass of the neutron star (NS) is susceptible
to $G_v$. When there is no repulsive force, the NS maximum mass is only about
$1.5M_\odot$, but it becomes larger than $2.0M_\odot$ when the vector coupling
constant is about half of the {attractive} scalar coupling constant.
Surprisingly, the presence of the quark matter does not affect the canonical
mass of NS ($1.4M_\odot$), so observing the canonical mass of NSs can provide
unique constraints to the EoS of hadronic matter at high densities.
