  Searches for stellar companions to hot Jupiters (HJs) have revealed that
planetary systems hosting a HJ are approximately three times more likely to
have a stellar companion with a semimajor axis between 50 and 2000 AU, compared
to field stars. This correlation suggests that HJ formation is affected by the
stellar binary companion. A potential model is high-eccentricity migration, in
which the binary companion induces high-eccentricity Lidov-Kozai (LK)
oscillations in the proto-HJ orbit, triggering orbital migration driven by
tides. A pitfall of this `binary-LK' model is that the observed stellar
binaries hosting HJs are typically too wide to produce HJs in sufficient
numbers, because of suppression by short-range forces. We propose a
modification to the binary-LK model in which there is a second giant planet
orbiting the proto-HJ at a semimajor axis of several tens of AU. Such
companions are currently hidden to observations, but their presence could be
manifested by a propagation of the perturbation of the stellar binary companion
inwards to the proto-HJ, thereby overcoming the barrier imposed by short-range
forces. Our model does not require the planetary companion orbit to be
eccentric and/or inclined with respect to the proto-HJ, but its semimajor axis
should lie in a specific range given the planetary mass and binary semimajor
axis, and the inclination with respect to the binary should be near $40^\circ$
or $140^\circ$. Our prediction for planetary companions to HJs in stellar
binaries should be testable by future observations.
