We demonstrate a sensitive method for the nonlinear optical characterization of micrometer long waveguides, and apply it to typical silicon-on-insulator nanowires and to hybrid plasmonic waveguides. We demonstrate that our method can detect extremely small nonlinear phase shifts, as low as 7.5·10<(-4) rad. The high sensitivity achieved imparts an advantage when investigating the nonlinear behavior of metallic structures as their short propagation distances complicates the task for conventional methods. Our results constitute the first experimental observation of χ((3)) nonlinearities in the hybrid plasmonic platform and is important to test claims of hybrid plasmonic structures as candidates for efficient nonlinear optical devices.