  Suppose $X$ is a smooth projective connected curve defined over an
algebraically closed field $k$ of characteristic $p>0$ and $B \subset X(k)$ is
a finite, possibly empty, set of points. The Newton polygon of a degree $p$
Galois cover of $X$ with branch locus $B$ depends on the ramification
invariants of the cover. When $X$ is ordinary, for every possible set of branch
points and ramification invariants, we prove that there exists such a cover
whose Newton polygon is minimal or close to minimal.
