Let $M$ be the moduli stack of ordinary but possibly nodal elliptic curves over the field $\overline{\mathbf{F}_p}$. Then $M$ has a $\mathbb{Z}_p^{\times}$-torsor over it, given by the moduli scheme of "trivialized" elliptic curves: that is, elliptic curves equipped with an isomorphism between the formal group and $\widehat{\mathbf{G}_m}$. A theorem of Igusa states that this cover (or rather, inverse system of covers) is connected.

Away from $p$-primary information, is this the universal cover of $M$, or is there an additional portion of the fundamental group? (In other words: what is the fundamental group of Katz's ring of $p$-adic modular forms?)