  We explore the implications of the recent radius determination of PSR
J0740+6620 by the NICER experiment combined with the neutron skin measurement
by the PREX-II experiment and the associated inference of the slope of symmetry
energy, for the structure of hybrid stars with a strong first-order phase
transition from nucleonic to quark matter. We combine a covariant
density-functional nucleonic equation of state (EOS) with a
constant-speed-of-sound EOS for quark matter. We show that the radius and tidal
deformability ranges obtained from GW170817 can be reconciled with the
implication of the PREX-II experiment if there is a phase transition to quark
matter in the low-mass compact star. In the high-mass segment, the EoS needs to
be stiff to comply with the large-radius inference for PSR J0740+6620 and
J0030+0451 with masses $M\simeq 2M_{\odot}$ and $M\simeq 1.4M_{\odot}$. We show
that twin stars are not excluded, but the mass and radius ranges (with $M \geq
M_\odot$) are restricted to narrow domains $\Delta M_{\rm twin} \lesssim 0.05
M_\odot$ and $\Delta R_{\rm twin} \sim 1.0$~km. We also show that the existence
of twin configurations is compatible with the light companion in the GW190814
event being a hybrid star in the case of values of the sound-speed square
$s=0.6$ and $s=1/3$.
