Absorption of laser light in plasma via electron-ion collision (inverse bremsstrahlung) is known to decrease with the laser intensity as I0-3/2 or with the electron temperature as Te-3/2 where Coulomb logarithm ln Λ = 0.5ln(1 + k2min/k2max) in the expression of electron-ion collision frequency vei is assumed to be independent of ponderomotive velocity v0 = E0/ω which is unjustified. Here k-1min = vth/max(ω, ωp), and k-1max = Z/v2th are maximum and minimum cut-off distances of the colliding electron from the ion, vth = √Te is its thermal velocity, ω, ωp are laser and plasma frequency. Earlier with a total velocity v = (v20 + v2th)1/2 dependent ln Λ(v) it was reported that vei and corresponding fractional laser absorption (α) initially increases with increasing intensity, reaches a maximum value, and then fall according to the conventional I0-3/2 scaling. This anomalous increase in vei and α may be objected due to an artifact introduced in ln Λ(v) through k-1min ∝ v. Here we show similar anomalous increase of vei and α versus I0 (in the low temperature and under-dense density regime) with quantum and classical kinetic models of vei without using ln Λ, but a proper choice of the total velocity dependent inverse cut-off length kmax-1 ∝ v2 (in classical case) or kmax ∝ v (in quantum case). For a given I0 < 5 × 1014Wcm-2, vei versus Te also exhibits so far unnoticed identical anomalous increase as vei versus Io, even if the conventional kmax ∝ v2th, or kmax ∝ vth is chosen. However, for higher Te > 15 eV, anomalous growth of vei and a disappear. The total velocity dependent kmax in kinetic models, as proposed here, may explain anomalous increase of a with I0 measured in some earlier laser-plasma experiments. This work may be important to understand collisional absorption in the under-dense pre-plasma region due to low intensity pre-pulses and amplified spontaneous emission (ASE) pedestal in the context of laser induced inertial confinement fusion.