  We study the stationary distribution of the (spread-out) $d$-dimensional
contact process from the point of view of site percolation. In this process,
vertices of $\mathbb{Z}^d$ can be healthy (state 0) or infected (state 1). With
rate one infected sites recover, and with rate $\lambda$ they transmit the
infection to some other vertex chosen uniformly within a ball of radius $R$.
The classical phase transition result for this process states that there is a
critical value $\lambda_c(R)$ such that the process has a non-trivial
stationary distribution if and only if $\lambda > \lambda_c(R)$. In
configurations sampled from this stationary distribution, we study
nearest-neighbor site percolation of the set of infected sites; the associated
percolation threshold is denoted $\lambda_p(R)$. We prove that $\lambda_p(R)$
converges to $1/(1-p_c)$ as $R$ tends to infinity, where $p_c$ is the threshold
for Bernoulli site percolation on $\mathbb{Z}^d$. As a consequence, we prove
that $\lambda_p(R) > \lambda_c(R)$ for large enough $R$, answering an open
question of Liggett and Steif in the spread-out case.
