The relay has nothing to do with how much voltage you get. It only regulates how much goes to the battery. So your problem would seem to be in the genny. Seen a few that were oil soaked(field coils) and that grounds the system and lowers the output. The armature could be burnt, or the brushes not sliding in the guides good enough to keep contact with the com bars. I would see what the output is at the A terminal, and if it is still sub 6 volts, remove and clean the guts, check the brushes in the guides, and the spring pressure on the brushes. If you have any arcing, and the com bars have spaces that are wider between the com bars, then the armature is burnt up. You can test the armature for grounding to frame, with a ohm meter. One to the shaft, or stackings, and the other to each individual com(utator) bar. If you ohm to 0 anywhere, the armature is bad. A out of round comutator bar, will cause the brushes to hop, and lose contact, so smoothness is important too. Not to mention loose bearings, or a ground in the brush guide plate(the positive brushes).
Check the service manual. You need to 'flash' the coils if it's been out of service for a while. You adjust output by moving the third brush. If the battery is low, you won't read high voltage until it is charged.
Assuming you are running a stock 3 brush generator.