The neutral wire in this circuit is taken from the receptacle and spliced to the white on the new cable, and to a pigtail that runs back to the receptacle. The hot is spliced to the black cable wire and a pigtail back to hot on the receptacle.
This diagram shows the wiring for a switch to control multiple receptacles. The source for the circuit is at the switch and the receptacles are wired using pigtail splices to make the connections.
The white wire on the switch loop is used to carry current from the source to the switch and it is marked with black tape or paint to label it as hot. The black on the switch loop runs from the top switch terminal to the top half of the receptacle. With this arrangement the top half of the duplex is controlled with the switch and the bottom half is always hot.
This diagram illustrates wiring for a circuit breaker with a built-in ground fault circuit interrupter. This 20 amp ground fault interrupter breaker is a form of gfci that can be installed at the circuit source. This kind of circuit is used for dishwashers, whirlpool spas and other locations where water contact is likely.