We study Galois covers of the projective line branched at three points with Galois group $G$. When such a cover is defined over a $p$-adic field, it is known to have potentially good reduction to characteristic $p$ if $p$ does not divide the order of $G$. We give a sufficient criterion for good reduction, even when $p$ does divide the order of $G$, so long as the $p$-Sylow subgroup of $G$ is cyclic and the absolute ramification index of a field of definition of the cover is small enough. This extends work of (and answers a question of) Raynaud. Our proof depends on working very explicitly with Kummer extensions of complete discrete valuation rings with imperfect residue fields.