The Magnus hierarchy has been used for almost a century to study one-relator groups. Taking a topological viewpoint, we refine the Magnus hierarchy. With this new tool, we characterise quasi-convex one-relator hierarchies in the sense of Wise. This new characterisation has several applications: we confirm a conjecture of Louder and Wilton on one-relator groups with negative immersions, we characterise hyperbolic one-relator groups with exceptional intersection, and answer a question of Baumslag’s on parafree one-relator groups. Finally, we introduce two new families of two-generator one-relator groups and prove that Gersten’s hyperbolicity conjecture is true for all one-relator groups if and only if it is true for these families.