  Let $\chi$ be a non-principal Dirichlet character modulo a prime $p$. Let
$q_1<q_2$ denote the two smallest prime non-residues of $\chi$. We give
explicit upper bounds on $q_2$ that improve upon all known results. We also
provide a good upper estimate on the product $q_1 q_2$ which has an upcoming
application to the study of norm-Euclidean Galois fields.
