  We study the enhanced diffusivity in the so called elephant random walk model
with stops (ERWS) by including symmetric random walk steps at small probability
$\epsilon$. At any $\epsilon > 0$, the large time behavior transitions from
sub-diffusive at $\epsilon = 0$ to diffusive in a wedge shaped parameter regime
where the diffusivity is strictly above that in the un-perturbed ERWS model in
the $\epsilon \downarrow 0$ limit. The perturbed ERWS model is shown to be
solvable with the first two moments and their asymptotics calculated exactly in
both one and two space dimensions. The model provides a discrete analytical
setting of the residual diffusion phenomenon known for the passive scalar
transport in chaotic flows (e.g. generated by time periodic cellular flows and
statistically sub-diffusive) as molecular diffusivity tends to zero.
