For the supercritical contact process on the hyper-cubic lattice startedfrom 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 $x \ne y \in \mathbb Z^d$, is stochastically tight. Second, for each $\varepsilon >0$ there existsxsuch that, for infinitely many integers $n$, $t(nx)< t((n+ 1)x)$ withprobability larger than $1-\varepsilon$. A key ingredient in our proofs is a tightness resultconcerning the essential hitting times of the supercritical contact process introduced by Garet and Marchand (2012)