This paper, jointly written with A. Deopurkar, confirms an expectation that the Picard groups of certain Hurwitz stacks of simply-branched, low-degree covers of the projective line are torsion. Perhaps the most interesting part of the paper concerns the investigation of the stratification of these Hurwitz stacks by the splitting type of natural vector bundles attached to covers. We establish some important dimension bounds on some of these strata, crucially identifying exactly when they are divisorial, and then use excision arguments to establish the result.