  By a result of Moonen, there are exactly 20 positive-dimensional families of
cyclic covers of the projective line for which the Torelli image is open and
dense in the associated Shimura variety. For each of these, we compute the
Newton polygons, and the $\mu$-ordinary Ekedahl--Oort type, occurring in the
characteristic $p$ reduction of the Shimura variety. We prove that all but a
few of the Newton polygons appear on the open Torelli locus. As an application,
we produce multiple new examples of Newton polygons and Ekedahl--Oort types of
Jacobians of smooth curves in characteristic $p$. Under certain congruence
conditions on $p$, these include: the supersingular Newton polygon for genus
$5,6,7$; fourteen new non-supersingular Newton polygons for genus $5-7$; eleven
new Ekedahl--Oort types for genus $4-7$ and, for all $g \geq 6$, the Newton
polygon with $p$-rank $g-6$ with slopes $1/6$ and $5/6$.
