  Let $f\colon C \rightarrow \mathbb{P}^1$ be a degree $k$ genus $g$ cover. The
stratification of line bundles $L \in \mathrm{Pic}^d(C)$ by the splitting type
of $f_*L$ is a refinement of the stratification by Brill-Noether loci
$W^r_d(C)$. We prove that for general degree $k$ covers, these strata are
smooth of the expected dimension. In particular, this determines the dimensions
of all irreducible components of $W^r_d(C)$ for a general $k$-gonal curve
(there are often components of different dimensions), extending results of
Pflueger and Jensen-Ranganathan. The results here apply over any algebraically
closed field.
