Can Six Sigma help the organization to accelerate or further improve this situation?
Good six sigma efforts (even 15 years ago) and lean share many of the same tools and principles that come from earlier TQM and such like efforts. There are some tools that are primarily associated with six sigma (like design of experiments). But those tools far precede "six sigma" even in their application in business. And those tools could certainly be useful in most lean organizations. There is no reason they couldn't just adopt those management tools.
Given that just in time was developed and made popular by Toyota and Deming long before the term six sigma was coined it certainly can be done expertly without six sigma tools. Six sigma tools can certainly help in my opinion, though.
I wouldn't weigh the benefit of any tools or methods or principles based on what category people places them in but instead I would build a management system based on the need of the organization. My preference is for Deming methods which form the foundation of lean and I also am a big fan of design of experiments (which most Deming and lean efforts do not use).
My father taught me design of experiments and Deming methods as a child and both have always made a great deal of sense to me. He wrote with George Box and Stu Hunter, what is seen by many as the premier design of experiments textbook, Statistics for Experimenters and he taught management improvement based on Deming's ideas, statistics, successful evidence based management principles... for decades.
These tools and methods all can be used together. A blog post of mine from 2005 on lean, six sigma, Deming, operational excellence and other management ideas.