Black hole-neutron star (BHNS) mergers are one of the most promising targets for multimessenger astronomy. Using general relativistic magnetohydrodynamic simulations of BHNS undergoing merger we previously showed that a magnetically driven jet can be launched by the disk+spinning black hole remnant if the neutron star is endowed with a dipole magnetic field extending from the interior into the exterior as in a radio pulsar. These self-consistent studies considered a BHNS system with mass ratio q=3:1, black hole spin a/MBH=0.75 aligned with the total orbital angular momentum, and a neutron star that is irrotational, threaded by an aligned magnetic field and modeled by an Γ-law equation of state with Γ=2. Here, as a crucial step in establishing BHNS systems as viable progenitors of central engines that power short gamma-ray bursts (sGRBs) and thereby solidify their role as multimessenger sources, we survey different BHNS configurations that differ in the spin of the BH companion (a/MBH=−0.5, 0, 0.5, 0.75), in the mass ratio (q=3:1 and q=5:1) and in the orientation of the magnetic field (aligned and tilted by 90° with respect to the orbital angular momentum). We find that by Δt∼3500M−4000M∼88(MNS/1.4  M⊙)  ms−100(MNS/1.4  M⊙)  ms after the peak gravitational wave signal a magnetically driven jet is launched in the cases where the initial spin of the BH companion is a/MBH=0.5 or 0.75. The lifetime of the jets [Δt∼0.5(MNS/1.4  M⊙)  s−0.7(MNS/1.4  M⊙)  s] and their outgoing Poynting luminosities [Ljet∼1051±1  erg/s] are consistent with typical sGRBs, as well as with the Blandford-Znajek mechanism for launching jets and their associated Poynting luminosities. By the time we terminate our simulations, we do not observe either an outflow or a large-scale magnetic field collimation in the other configurations we simulate. These results suggest that future multimessenger detections from BHNSs are more likely produced by binaries with highly spinning BH companions and small tilt-angle magnetic fields, though other physical processes not considered here, such as neutrino annihilation, may help to power jets in general cases.