Adding an outlet to a circuit
Easy peasy.   

The black wires in the back of the box are certainly always-hot.   

The white wires are certainly neutral, and they are using the receptacle to be a splice (i.e. To replace a wire nut) while also supplying the receptacle.  

The red is your switched hot from the switch, which you defeated.  

You want to add a third hot and neutral from your new cable.  It would also be nice to power the receptacle straight off the always-hot (black) and abandon that extra loop up to the switch.  It would also be nice to stop using those unreliable and troublesome "back stab" wire connections.   

You can't use two wiring methods (backstab and screw) on the same receptacle, so you can't splice 3 wires to a receptacle.  The exception is the Leviton "screw-and-clamp" types, which support 4 wires per side and are about $3.00.  Alternately you could pigtail the receptacle.  

So I would add the new cable.   Join the now-3 black wires on the brass side of the Leviton screw-and-clamp, or the 4 wires (with pigtail from the receptacle) with a red wire nut.   Same thing with the whites. 

Red wire nuts are ideal to bind four #12s or #14s, it helps to tape any 2 wires together so you are trying to line up 2 bound pairs of wires instead of four loose ones.

As for the red, I would splice the existing and your new red together with a yellow wire nut. Cap the red wire off in the new receptacle, and also go back into the switch box and cap it off there.  Now it's de-energized, but it's in the wall in case the next homeowner wants to set up a switched outlet at either old or your new location.