Dense suspensions of hard particles in a Newtonian liquid can be jammed by shear when the applied stress exceeds a certain threshold. However, this jamming transition from a fluid into a solidified state cannot be probed with conventional steady-state rheology because the stress distribution inside the material cannot be controlled with sufficient precision. Here we introduce and validate a method that overcomes this obstacle. Rapidly propagating shear fronts are generated and used to establish well-controlled local stress conditions that sweep across the material. Exploiting such transient flows, we can track how a dense suspension approaches its shear-jammed state dynamically and quantitatively map out the onset stress for solidification in a state diagram.