In this work, we re-examine collisional absorption of 800 nm wavelength laser pulses in under-dense plasma. For a given temperature and density of the plasma, most of the conventional models of the electron-ion collision frequency νei, with a Coulomb logarithm independent of the electron-ponderomotive velocity, show that νei and the corresponding fractional laser absorption α remain almost constant (or decrease slowly) up to a value Ic of the peak intensity I0 of the laser pulse, and then νei and α decrease as ≈I0−3/2 when I0 is increased beyond Ic. On the contrary, below some temperature (≲10 eV) and density, with a total-velocity (thermal velocity plus the ponderomotive velocity) dependent Coulomb logarithm, we find that νei and α grow hand in hand up to a maximum value around Ic followed by the conventional I0−3/2 decrease when I0>Ic. Such a non-conventional anomalous variation of α with I0 was observed in some earlier experiments, but no explanation has been given so far. The modified Coulomb logarithm considered in this work may be responsible for those experimental observations. With increasing temperature and density, the anomalous behavior is found to disappear even with the modified Coulomb logarithm, and the variation of νei and α with I0 approach to the conventional scenario.