At temperatures below the nominal glass transition temperature T-g omega, the structural a-relaxation and the Johari-Goldstein (JG) beta-relaxation are too slow to contribute to susceptibility measured at frequencies higher than 1 GHz. This is particularly clear in the neighborhood of the secondary glass transition temperature T-g beta, which can be obtained directly by positronium annihilation lifetime spectroscopy (PALS) and adiabatic calorimetry, or deduced from the temperature at which the JG beta-relaxation time tau(beta) reaches 1000 s. The fast process at such high frequencies comes from the vibrations and caged molecules dynamics manifested as the nearly constant loss (NCL) in susceptibility measurements, elastic scattering intensity, I(Q, T), or the mean-square-displacement, < u(2)(T)>, in quasielastic neutron scattering experiment. Remarkably, we find for many different glass-formers that the NCL, I, or < u(2 >) measurednear To. In paper I (Capaccioli, S.; et al. J. Phys. Chem. B 2015, 119 (28), 8800-8808) we have made known this property in the case of the polyalcohols and a pharmaceutical glass former, flufenamic acid studied by THz dielectric spectroscopy, and explained it by the coupling of the NCL to the JG beta-relaxation, and the density dependence of these processes. In this paper II, we extend the consideration of the high frequency response to broader range from 100 MHz to THz in the glassy state of many polymers observed by quasielastic light scattering, Brillouin scattering, quasielastic neutron scattering, and GHz-THz dielectric relaxation. In all cases, the NCL changes its T-dependence at some temperature, T-HF, below T-ga, which is approximately the same as T-g beta. The latter is independently determined by PALS, or adiabatic calorimetry, or low frequency dielectric and mechanical spectroscopy. The property, T-HF approximate to T-g beta, had not been pointed out before by others or in any of the quasielastic neutron and light scattering studies of various amorphous polymers and van der Waals small molecular glass-formers over the past three decades. The generality and fundamental importance of this novel property revitalize the data from these previous publications, making it necessary to be reckoned with in any attempt to solve the glass transition problem. In our rationalization, the property arises first from the fact that the JG beta-relaxation and the caged dynamics both depends on density and entropy. Second, the JG beta-relaxation is the terminator of the caged dynamics, and hence the two processes are inseparable or effectively coupled. Consequently, the occurrence of the secondary glass transition at T-g beta necessarily is accompanied by corresponding change in the temperature dependence of the NCL, I, or < u(2 >) of the fast caged dynamics at THF approximate to Tg(beta).