Low-frequency kinetic Alfven waves (KAWs) are studied as a possible source for the strong heating of ions across the magnetic field in the solar corona. It is shown that test ions moving in the electromagnetic fields of KAWs undergo an increase in their cross-field energy because of the superadiabatic acceleration in the vicinity of the demagnetizing wave phases. In particular, it is found that KAW wave trains, with a transversal wavelength of the order of 40 proton gyroradii and with a peak wave/background magnetic field ratio greater than or similar to0.1, increase the cross-field energy of O5+ oxygen ions by 1-2 orders. The required short perpendicular wavelengths can be produced by the phase mixing of MHD Alfven waves, propagating upward from the coronal base. The superadiabatic acceleration provides an alternative to the ion-cyclotron explanation for the intense transverse heating of O+5 and Mg9+ ions observed by the Solar and Heliospheric Observatory at 1.5-3 solar radii.