  We consider a one-dimensional search process under stochastic resetting
conditions. A target is located at $b\geq0$ and a searcher, starting from the
origin, performs a discrete-time random walk with independent jumps drawn from
a heavy-tailed distribution. Before each jump, there is a given probability $r$
of restarting the walk from the initial position. The efficiency of a "myopic
search" - in which the search stops upon crossing the target for the first time
- is usually characterized in terms of the first-passage time $\tau$. On the
other hand, great relevance is encapsulated by the leapover length $l =
x_{\tau} - b$, which measures how far from the target the search ends. For
symmetric heavy-tailed jump distributions, in the absence of resetting the
average leapover is always infinite. Here we show instead that resetting
induces a finite average leapover $\ell_b(r)$ if the mean jump length is
finite. We compute exactly $\ell_b(r)$ and determine the condition under which
resetting allows for nontrivial optimization, i.e., for the existence of $r^*$
such that $\ell_b(r^*)$ is minimal and smaller than the average leapover of the
single jump.
