There are plenty of ways to get this done smoothly. The common element is to attach a "fish" or retrieval line to one end of the cable BEFORE removing it from the frame.

In some cases I employ a short cut. After getting some slack to work with, I cut the cable at the lever end so I have plain wire there. I use shrink tubing to splice the new wire to the old and push/pull it into place. It's basically the same, but less back and forth.

BTW - the other key element of the job is to remember to thread all the ferrules onto the cable, and thread the housing through any fitting it might route through. Otherwise, you'll get extra practice for this kind of work.