Best practice would be to run a new wire. Your ideas of using the water pipe, even with another rod, won't work unless you bond back to the existing ground electrode system.
You could share the EGC of the new receptacle circuit you have run but you do have to worry about protecting the wire you run. I would ask if your AHJ is OK with just stringing a THHN over to the receptacle box
250.120(C) Equipment Grounding Conductors Smaller Than 6 AWG. Equipment grounding conductors smaller than 6 AWG shall be protected from physical damage by a raceway or cable armor except where run in hollow spaces of walls or partitions, where not subject to physical damage, or where protected from physical damage.