The stability of a hierarchical triple star may be decided by a simple criterion, which was derived in Paper I. However, there is a region in the phase space where the stability limit Q_max is raised by a factor of two in a small region of the phase space with respect to the surrounding phase space. The phase space is defined by the inner and outer eccentricities e_in and e_out, respectively, as well as by the inclination ι_tot between the inner and outer orbits. Additional parameters of the phase space are the masses of the three bodies. We study by numerical integration the orbits of over 100 000 triple systems in the resonance region. We find that the instability that causes the high value of Q_max arises from the octupole Kozai-Lidov resonance. This resonance region has rather equal contributions from the quadrupole and octupole terms and leads to secular evolution with an amplitude larger than either of the two oscillations in isolation. The conditions for this situation are best satisfied near the relative inclination ι = 140°. Additionally, the relative orientation of the two orbits plays a decisive role: the resonance is found only at certain values of the orbit's node line longitude Ω. An analytical approximation of the energy change in a single close encounter between the inner and outer systems suggests a cos 2Ω dependence of Q_max on Ω, which seems to be qualitatively valid. We model Q_max as a function of cos ι and e_in by a Gaussian function.