You want the thickest and widest floss you can find. My thickest floss is .1mm (.004"). It's the one in the photo. If you use two pieces, it surrounds most of the rod, which is ideal but not necessary.
Unfortunately, floss packages never tell you the thickness. So beg, borrow, steal or buy a variety and you'll have a good selection. Ask your dentist for samples. Maybe people you know have some that is too thick for them. I keep four different packs handy, and that's all I need to find the right size, in some cases doubling it. Remember to add oil before inserting the rod for testing or assembly.
You are describing a large amount of end-play. Maybe the floss that works inside the tube will not be sufficient for the end-play. A way to take up end-play is to wind some polyester sewing thread tightly in the gap, tie a half-knot at least, then secure the knot with a precise droplet of thin CA glue. Keep the thread free of oil so the CA can absorb and harden. The thread forms a durable spacer, if the glue secures the knot. I just revisited an old repair I did, and found an old thread spacer that was so invisible, I had forgotten that it was there. It held up fine. If anyone knows where to get really small washers that can do this better, please write in!
