The recent discovery by TESS of the short-period giant planet WD1856+534b is a major milestone in the emerging field of white-dwarf (WD) planetary science. The planet and its host belong to an hierarchical quadruple system with two M-dwarf companions. Secular interactions between the planet and the companion stars through the Lidov-Kozai (LK) effect may have caused high-eccentricity migration of the planet to its current location. We show that a secular inclination resonance between the planet and the companion stars greatly enhances the probability of LK migration in this system by broadening the window of initial inclinations for which extreme eccentricity excitation is possible. This resonance can remain inactive until the host star loses mass during the AGB phase, thereby delaying migration until the WD phase. Octupole effects can also expand the “extreme” LK window. We suggest that WD1856+534b was likely an ordinary “cold Jupiter” during the host’s main sequence phase, with a semi-major axis of ~10-20 AU. However, if this system is typical, it implies a large population of short-period giant planets around WDs, difficult to explain through high-e migration alone.