  Let $\varphi: C\to \mathbb{P}^1$ be a degree $d$ cover of curves. In work by
Castryck, Zhao and the author, we showed how one can attach to each partition
$\lambda$ of $d$ a multi-set of scrollar invariants of $\lambda$ with respect
to $\varphi$. We studied these invariants when $\varphi$ is simply branched,
and related these new scrollar invariants to known geometric data. In this
article we show how one can remove this simple branching condition, using the
notion of $S_d$-closure as developed by Bhargava and Satriano. With this new
framework, we are able to generalize all results from this work to arbitrary
covers $C\to \mathbb{P}^1$. In particular, we obtain a syzygy-free
interpretation for the splitting types of the syzygy bundles in the
Casnati--Ekedahl resolution of an arbitrary cover $\varphi: C\to \mathbb{P}^1$.
By using the Maroni bound, we are able to give new general bounds on these
splitting types.
