  The hadron-quark phase transition in core-collapse supernovae (CCSNe) has the
potential to trigger explosions in otherwise nonexploding models. However,
those hybrid supernova equations of state (EOS) shown to trigger an explosion
do not support the observational 2 M$_\odot$ neutron star maximum mass
constraint. In this work, we analyze cold hybrid stars by the means of a
systematic parameter scan for the phase transition properties, with the aim to
develop a new hybrid supernova EOS. The hadronic phase is described with the
state-of-the-art supernova EOS HS(DD2), and quark matter by an EOS with a
constant speed of sound (CSS) of $c_{QM}^2=1/3$. We find promising cases which
meet the 2 M$_\odot$ criterion and are interesting for CCSN explosions. We show
that the very simple CSS EOS is transferable into the well-known thermodynamic
bag model, important for future application in CCSN simulations. In the second
part, the occurrence of reconfinement and multiple phase transitions is
discussed. In the last part, the influence of hyperons in our parameter scan is
studied. Including hyperons no change in the general behavior is found, except
for overall lower maximum masses. In both cases (with and without hyperons) we
find that quark matter with $c_{QM}^2=1/3$ can increase the maximum mass only
if reconfinement is suppressed or if quark matter is absolutely stable.
