  For the supercritical contact process on the hyper-cubic lattice started from
a single infection at the origin and conditioned on survival, we establish two
uniformity results for the hitting times $t(x)$, defined for each site $x$ as
the first time at which it becomes infected. First, the family of random
variables $(t(x)-t(y))/|x-y|$, indexed by $x \neq y$ in $\mathbb{Z}^d$, is
stochastically tight. Second, for each $\varepsilon >0$ there exists $x$ such
that, for infinitely many integers $n$, $t(nx) < t((n+1)x)$ with probability
larger than $1-\varepsilon$. A key ingredient in our proofs is a tightness
result concerning the essential hitting times of the supercritical contact
process introduced by Garet and Marchand (Ann.\ Appl.\ Probab., 2012).
