It sounds like you have a switched outlet, perhaps with the switch disabled. The best way to accomplish adding an outlet would be to remove the black from the existing outlet and splice it to the black to the new outlet plus a short, 6" jumper wire. Put the jumper back to the outlet where the existing black wire was. Attach the white to the unused terminal with the other white on the outlet and splice the ground to the ground going to the existing outlet, again with a short jumper so it can be re-attached back where it was.
If you find the switch turning that outlet off, try changing the black to the red wire on the existing outlet.
If the existing outlet is indeed switched, the small tab between the red and black wires will have been removed. If you only remove the red and attach the new black where it was, it will not work - by splicing to one of the existing wires and maintaining the terminations, everything will work.