  We extend the quark mean-field (QMF) model for nuclear matter and study the
possible presence of quark matter inside the cores of neutron stars. A sharp
first-order hadron-quark phase transition is implemented combining the QMF for
the hadronic phase with "constant-speed-of-sound" parametrization for the
high-density quark phase. The interplay of the nuclear symmetry energy slope
parameter, $L$, and the dimensionless phase transition parameters (the
transition density $n_{\rm trans}/n_0$, the transition strength
$\Delta\varepsilon/\varepsilon_{\rm trans}$, and the sound speed squared in
quark matter $c^2_{\rm QM}$) are then systematically explored for the hybrid
star proprieties, especially the maximum mass $M_{\rm max}$ and the radius and
the tidal deformability of a typical $1.4 \,M_{\odot}$ star. We show the strong
correlation between the symmetry energy slope $L$ and the typical stellar
radius $R_{1.4}$, similar to that previously found for neutron stars without a
phase transition. With the inclusion of phase transition, we obtain robust
limits on the maximum mass ($M_{\rm max}< 3.6 \,M_{\odot}$) and the radius of
$1.4 \,M_{\odot}$ stars ($R_{1.4}\gtrsim 9.6~\rm km$), and we find that a
too-weak ($\Delta\varepsilon/\varepsilon_{\rm trans}\lesssim 0.2$) phase
transition taking place at low densities $\lesssim 1.3-1.5 \, n_0$ is strongly
disfavored. We also demonstrate that future measurements of the radius and
tidal deformability of $\sim 1.4 \,M_{\odot}$ stars, as well as the mass
measurement of very massive pulsars, can help reveal the presence and amount of
quark matter in compact objects.
