How this goes depends on the age of the packing and its composition. If it's old crap, all of it will require replacement. Relatively new, maybe be able to get away with adding a ring, if there's any room.
If you are going to replace all of the packing, this will take several hours or more. In order to keep the incoming water under control and your sox dry while digging out the old packing, try this :
Once you have the incoming water stopped at the shaft log, you have all day to do the job with no water ingress except for a small amount when pressing the new packing into place.
Practice just getting the string into place ON and OFF to start with.
This is how I do mine in the water as I'm not about to do it on the hard where I'm paying $40.00/day. I stick to the stuff that really can't be done in the water.