A previous conjecture by the authors about a new regime of Arnold diffusion with a power-law dependence of the diffusion rate on perturbation strength is confirmed by detailed theoretical evaluation. A new effect of slow (logarithmic) dependence of the power-law exponent on the perturbation parameter is conjectured. The theory developed seems to allow for a new interpretation of the recent extensive numerical experiments on Arnold diffusion in a particular many-dimensional model of Kaneko and Konishi even in the presence of some global chaos.