Atomic hydrogen (H i) serves a crucial role in connecting galactic-scale properties such as star formation with the large-scale structure of the Universe. While recent numerical simulations have successfully matched the observed covering fraction of H i near Lyman Break Galaxies (LBGs) and in the foreground of luminous quasars at redshifts z less than or similar to 3, the low-mass end remains as-of-yet unexplored in observational and computational surveys. We employ a cosmological, hydrodynamical simulation (FIREbox) supplemented with zoom-in simulations (MassiveFIRE) from the Feedback In Realistic Environments (FIRE) project to investigate the H i covering fraction of Lyman Limit Systems (NHI greater than or similar to 101(7.2) cm(-2)) across a wide range of redshifts (z=0-6) and halo masses (10(8)-10(13)M(circle dot) at z=0, 10(8)-10(11)M(circle dot) at z=6) in the absence of feedback from active galactic nuclei. We find that the covering fraction inside haloes exhibits a strong increase with redshift, with only a weak dependence on halo mass for higher mass haloes. For massive haloes (M-vir similar to 10(11)-10(12)M(circle dot)), the radial profiles showcase scale-invariance and remain independent of mass. The radial dependence is well captured by a fitting function. The covering fractions in our simulations are in good agreement with measurements of the covering fraction in LBGs. Our comprehensive analysis unveils a complex dependence with redshift and halo mass for haloes with M-vir less than or similar to 10(10)M(circle dot) that future observations aim to constrain, providing key insights into the physics of structure formation and gas assembly.