We give new criteria for ballistic behavior of random walks in random environment which are perturbations of the simple symmetric random walk on $\mathbb Z^d$ in dimensions $d\ge 4$. Our results extend those of Sznitman [Ann. Probab. 31, no. 1, 285-322 (2003)] and the recent ones of Ramírez and Saglietti [Preprint, arXiv:1808.01523], and allow us to exhibit new examples in dimensions $d\ge 4$ of ballistic random walks which do not satisfy Kalikow's condition. Our criteria implies ballisticity whenever the average of the local drift of the walk is not too small compared with an appropriate moment of the centered environment. The proof relies on a concentration inequality of Boucheron et al. [Ann. Probab. 33, no. 2, 514-560 (2005)].