  Several stars detected moving at velocities near to or exceeding the Galactic
escape speed likely originated in the Milky Way disc. We quantitatively explore
the `binary supernova scenario' hypothesis, wherein these `hyper-runaway' stars
are ejected at large peculiar velocities when their close, massive binary
companions undergo a core-collapse supernova and the binary is disrupted. We
perform an extensive suite of binary population synthesis simulations evolving
massive systems to determine the assumptions and parameters which most impact
the ejection rate of fast stars. In a simulation tailored to eject fast stars,
we find the most likely hyper-runaway star progenitor binary is composed of a
massive ($\sim$$30\,\mathrm{M_{\odot}}$) primary and a
$\sim$$3-4\,\mathrm{M_{\odot}}$ companion on an orbital period that shrinks to
$\lesssim$1 day prior to the core collapse following a common envelope phase.
The black hole remnant formed from the primary must receive a natal kick
$\gtrsim$1000 $\mathrm{km\ s^{-1}}$ to disrupt the binary and eject the
companion at a large velocity. We compare the fast stars produced in these
simulations to a contemporary census of early-type Milky Way hyper-runaway star
candidates. We find that these rare objects may be produced in sufficient
number only when poorly-constrained binary evolution parameters related to the
strength of post-core collapse remnant natal kicks and common envelope
efficiency are adjusted to values currently unsupported -- but not excluded --
by the literature. We discuss observational implications that may constrain the
existence of these putative progenitor systems.
