Supercritical Nonlinear Schr\"odinger Equations II: Almost Global Existence

We prove almost global existence for supercritical nonlinear Schr"odinger equations on the $d$-torus ($d$ arbitrary) on the good geometry selected in part I. This is seen as the Cauchy consequence of I, since the known invariant measure of smooth solutions are supported on KAM tori. In the high frequency limit, these quantitative solutions could also be relevant to Cauchy problems for compressible Euler equations.