  We sharpen ellipticity criteria for random walks in i.i.d. random
environments introduced by Campos and Ram\'{\i}rez which ensure ballistic
behavior. Furthermore, we construct new examples of random environments for
which the walk satisfies the polynomial ballisticity criteria of Berger,
Drewitz and Ram\'{\i}rez. As a corollary, we can exhibit a new range of values
for the parameters of Dirichlet random environments in dimension $d=2$ under
which the corresponding random walk is ballistic.
