The molecular gas content of normal galaxies at z> 4 is poorly constrained because the commonly used molecular gas tracers become hard to detect at these high redshifts. We use the [CII] 158 mu m luminosity, which was recently proposed as a molecular gas tracer, to estimate the molecular gas content in a large sample of main sequence star-forming galaxies at z=4.4-5.9, with a median stellar mass of 10(9.7) M-circle dot, drawn from the ALMA Large Program to INvestigate [CII] at Early times survey. The agreement between the molecular gas masses derived from [CII] luminosities, dynamical masses, and rest-frame 850 mu m luminosities extrapolated from the rest-frame 158 mu m continuum supports [CII] as a reliable tracer of molecular gas in our sample. We find a continuous decline of the molecular gas depletion timescale from z=0 to z=5.9, which reaches a mean value of (4.6 +/- 0.8) x 10(8) yr at z similar to 5.5, only a factor of between two and three shorter than in present-day galaxies. This suggests a mild enhancement of the star formation efficiency toward high redshifts. Our estimates also show that the previously reported rise in the molecular gas fraction flattens off above z similar to 3.7 to achieve a mean value of 63%+/- 3% over z=4.4-5.9. This redshift evolution of the gas fraction is in line with that of the specific star formation rate. We use multi-epoch abundance-matching to follow the gas fraction evolution across cosmic time of progenitors of z=0 Milky Way-like galaxies in similar to 10(13) M-circle dot halos and of more massive z=0 galaxies in similar to 10(14) M-circle dot halos. Interestingly, the former progenitors show a monotonic increase of the gas fraction with redshift, while the latter show a steep rise from z=0 to z similar to 2 followed by a constant gas fraction from z similar to 2 to z=5.9. We discuss three possible effects, namely outflows, a pause in gas supply, and over-efficient star formation, which may jointly contribute to the gas fraction plateau of the latter massive galaxies.