Adding hot receptacle from current light switch power?
Given that the line cable (hot and neutral) enters the switch box, you can run a new cable from the light switch box to a new receptacle and connect it so the switch will still switch the existing receptacle.

You cannot run a new cable from the existing switched receptacle to get an always on receptacle.   

If you wanted to do away with the switched receptacle, you could remove the switch from the circuit and connect the wires with a wire nut. The existing receptacle would then be always on.

You could install a ceiling box and light fixture and run a cable from the switch box to the light fixture. That way the switch would control the ceiling light.