This paper addresses velocity-level redundancy resolution for S-R-S type redundant manipulators, aiming at precise path following in the presence of kinematic singularities. A novel joint-velocity decomposition scheme that complements and avoids some drawbacks of a previous study based on the Singularity-Consistent method is introduced. As a result, it becomes possible to follow almost any singular path in workspace, including paths passing through bifurcations (nondegenerate singular paths). Further on, the algorithmic singularity introduced in the previous study (the “inner obstacle”) disappears in the new formulation. The problem of wrist singularity is also addressed and solved via a proper self motion component. Only a limited subset of singular paths cannot be handled with the method: those that pass through the double elbow-wrist singularity (an isolated point at the workspace boundary). The effectiveness of the proposed method is illustrated via numerical simulations.