As cheap as the packing coils are you might get a pack of 1/8 and a pack of 3/16. When you back off of the compression fitting, you will get some water intrussion, but not much. You will be able to feel the spacing that the packing will fit into. Should be able to determine the largest that can be tapped into place. You will cut the new coil to just go around the shaft with a slanted cut to provide and over lap. If you take out whats in there you need 3 pieces, but if you just add, one or two will do. Hardest part of this is to loosen the jam nut if the joint has been there for several years.
I've done this several times. No big deal. You may want to tighten up for one drip every 15 sec, then retighten after you have run the shaft a while. Good luck. Phil in FL