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 $z \lesssim 3$, 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 ($N_{{\text{H}}\, \rm{{\small I}}} \gtrsim 10^{17.2}$ cm$^{-2}$) across a wide range of redshifts ($z=0-6$) and halo masses ($10^8-10^{13} \, \,\mathrm{ M}_{\odot }$ at $z=0$, $10^8-10^{11}\, \,\mathrm{ M}_{\odot }$ 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 ($M_{\mathrm{vir}} \sim 10^{11}-10^{12} \,\mathrm{ M}_{\odot }$), 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 $M_{\mathrm{vir}} \lesssim 10^{10} \,\mathrm{ M}_{\odot }$ that future observations aim to constrain, providing key insights into the physics of structure formation and gas assembly.