Maximally random jammed states of hard spheres are prototypical glasses. We study the small wavenumber k behavior of the structure factor S(k) of overcompressed million-sphere packings as a function of density up to the jammed state. We find both a precursor to the glassy jammed state evident long before the jamming density is reached and two associated growing length scales, one extracted from the volume integral of the direct correlation function c(r) and the other from the small-k behavior of the structure factor S(k), that can diverge at the "critical" jammed state. We also define a nonequilibrium index X and use it to demonstrate that the packings studied are intrinsically nonequilibrium in nature well before the critical state is reached. The extension of the results reported in the present work to supercooled atomic-liquid models in which the atoms interact with both repulsive and attractive forces is also discussed.