  In independent bond percolation on $\mathbb{Z}^d$ with parameter $p$, if one
removes the vertices of the infinite cluster (and incident edges), for which
values of $p$ does the remaining graph contain an infinite cluster?
Grimmett-Holroyd-Kozma used the triangle condition to show that for $d \geq
19$, the set of such $p$ contains values strictly larger than the percolation
threshold $p_c$. With the work of Fitzner-van der Hofstad, this has been
reduced to $d \geq 11$. We improve this result by showing that for $d \geq 10$
and some $p>p_c$, there are infinite paths consisting of "shielded" vertices
--- vertices all whose adjacent edges are closed --- which must be in the
complement of the infinite cluster. Using numerical values of $p_c$, this bound
can be reduced to $d \geq 7$. Our methods are elementary and do not require the
triangle condition.
