A stuffing box should not just be blindly tightened and if your running out of room, that makes me a bit uneasy. Pulling out stuffing and replacing is usually an easy job, safely accomplished in the water. If you run into issues, just stuff a little anything in there while you go to the chandler or hardware store. A rag, some paper towels. If the shaft is not turning, just about anything is going to work.
Best to get new packing in there. Come prepared with a couple corkscrew tools. If the packing is baked and hardened from being overtightened, it can be a bit crispy to pull out and it is possible to break a tool. Packing that is old and not hardened will usually come out quite easily. Another hint, if you guess wrong at the perfect packing size, just massage it with a mallet/hammer gently to make it a little thicker, or thinner. You can make any close size work and it will simply cause the overall depth to change just a bit.
I’m not at all a fan of adding a new ring to old packing.
Personally, I prefer gfo.
If you have troubles getting new packing to seal without over-tightening, then inspect the shaft for scoring. That’s a real possibility resulting from prior over tightening. If the shaft is in good shape, and you put in the new rings at the right shape and size, you should tighten gently at first and creep up on the right adjustment slowly while running a little at a time. If you are starting to get tight and not seeing obvious reduction in dripping, instead of continuing to get tighter and tighter, check whether the shaft is wobbling and the running gear needs alignment. You won’t be able to get a stuffing box to perform right unless the running gear is in alignment and otherwise good shape. This ultimately is what often leads to over tightening and baked packing.