We describe a general Monte Carlo (MC) suite for the efficient relaxation of dense systems of freely jointed chains of hard spheres for packing densities (phi) ranging from dilute ones (phi approximate to 0.1) up to the vicinity of the maximally random jammed (MRJ) state (phi approximate to 0.639). Key components of the new MC scheme are as follows: (i) modified versions of state-of-the-art chain-connectivity altering moves tailored to function efficiently even for very dense random chain assemblies and (ii) localized MC moves, all executed in an adaptive configurational-bias pattern. Two different athermal systems were simulated consisting of 100 and 50 chains with average lengths N-av = 12 and 24, respectively, over a wide range of packing densities. In all cases studied, computational efficiency is compared against corresponding results obtained from extensive applications of conventional MC techniques; it is shown that the proposed MC scheme outperforms by several orders of magnitude all existing methods in relaxing the short- and long-range characteristics of the simulated systems and is thus the only available vehicle to generate and equilibrate (and successively characterize) representative model chain configurations near the MRJ state. Structural results, obtained from averaging over very long MC trajectories, show the dependence of bonded geometry (quantified by the bending and torsion angle distributions), chain dimensions (quantified by the end-to-end distance and radius of gyration) and local packing on volume fraction.