Global redundancy resolution (GRR) roadmaps is a novel concept in robotics that facilitates the mapping from task space paths to configuration space paths in a legible, predictable, and repeatable way. Such roadmaps could find widespread utility in applications such as safe teleoperation, consistent path planning, and motion primitives generation. However, previous methods to compute GRR roadmaps often necessitate a lengthy computation time and produce non-smooth paths, limiting their practical efficacy. To address this challenge, we introduce a novel method Expansion-GRR that leverages efficient configuration space projections and enables rapid generation of smooth roadmaps that satisfy the task constraints. Additionally, we propose a simple multi-seed strategy that further enhances the final quality. We conducted experiments in simulation with a 5-link planar manipulator and a Kinova arm. We were able to generate the Expansion-GRR roadmaps up to 2 orders of magnitude faster while achieving higher smoothness. We also demonstrate the utility of the GRR roadmaps in teleoperation tasks where our method outperformed prior methods and reactive IK solvers in terms of success rate and solution quality.