Current adversarial attacks for multi-class classifiers choose the target class for a given input naively, based on the classifier's confidence levels for various target classes. We present a novel adversarial targeting method, \textit{MALT - Mesoscopic Almost Linearity Targeting}, based on medium-scale almost linearity assumptions. Our attack wins over the current state of the art AutoAttack on the standard benchmark datasets CIFAR-100 and ImageNet and for a variety of robust models. In particular, our attack is \emph{five times faster} than AutoAttack, while successfully matching all of AutoAttack's successes and attacking additional samples that were previously out of reach. We then prove formally and demonstrate empirically that our targeting method, although inspired by linear predictors, also applies to standard non-linear models.