That switch box contains 2 hots and no neutral. So without running any new wire, your only option would be to re-wire the outlet so that it is no longer switched and use the switch box as a junction box.
Here's a before and after diagram. Note that the "neutral" between the switch and the outlet is actually your white wire that has been relabeled hot.
Thinking about it again, you could make your NEW outlet switched in this configuration.
Another alternative is to disconnect the existing supply, and run a new line to the switch/receptacle. It really just depends on which line is easier to run.
(Note, I just copied the original image, but didn't erase the ground. The ground should be connected to the box and the ground nut on the receptacle, and switch if it has one. At no point here should the ground be connected to neutral, or anything but a box or a ground screw. Thanks @Skaperen)
You should disconnect the other wire though, at both ends, and connect the conductors together so if anyone mistakenly energizes that line in the future, it will blow the breaker.
Best Answer
Perfectly by the book.
Neutral is available in the switch in case you ever want to replace it with a smart switch that needs power.
Black is always live and red is the switched live.