There's an alternative, actually--run the wire from the new outlet to the existing outlet. On that wire, strip the ends, but then also strip a 1" section about 5-6" from the end, so your wire kinda looks like this: (comes out of wall) =========------==============------ (end)
the === are insulated sections, the ---- are uninsulated sections. Do it for both the line (usually black) and neutral (white) wires.
Disconnect the right-hand outlet from the current-left-hand outlet. Use the uninsulated section in the middle of the new wire to wrap around the screws on the old left-hand outlet. Then use a wire nut to connect the ends of these wires to the ends of the wires coming from the right-hand outlet. You'll still need wire nuts, but you'll only have to put two wires into each.
You didn't mention anything about grounds, but make sure those are connected as well.
Thanks, Chris22, for providing a drawing to riff off.