In the first two papers separately on the polyalcohols and amorphous polymers of this series, we demonstrated that the fast dynamics observed in the glassy state at high frequencies above circa 1 GHz is the caged dynamics. We showed generally the intensity of the fast caged dynamics changes temperature dependence at a temperature T-HF nearly coincident with the secondary glass transition temperature To lower than the nominal glass transition temperature T-ga. The phenomenon is remarkable, since T-HF is determined from measurements of fast caged dynamics at short time scales typically in the ns to ps range, while To characterizes the secondary glass transition at which the Johari-Goldstein (JG) beta-relaxation time tau(JG) reaches a long time of similar to 10(3) s, determined directly either by positronium annihilation lifetime spectroscopy, calorimetry, or low frequency dielectric and mechanical relaxation spectroscopy. The existence of the secondary glass transition originates from the dependence of tau(JG) on density, previously proven by experiments performed at elevated pressure. The fact that T-HF approximate to T-g beta reflects the density dependence of the caged dynamics and coupling to the JG beta-relaxation. The generality of the phenomenon and its theoretical rationalization implies the same should be observable in other classes of glass-formers. In this paper, III, we consider two archetypal small molecular van der Waals glass-formers, ortho-terphenyl and toluene. The experimental data show the same phenomenon. The present paper extends the generality of the phenomenon and explanation from the polyalcohols, a pharmaceutical, and many polymers to the small molecular van der Waals glass-formers.