Lighting, Light Fixtures, Ceiling and Exhaust Fans - New ceiling light (using existing switch)

Found a few threads but none the same as mine so I'll post a new one. If it's a dupe, please point me to the existing thread...

I have an existing switch in a bedroom that controls one half of an outlet. Want to add a light fixture to the ceiling. The power goes to the outlet with one cable going to the switch (2 wire +ground) both black and white connect to the switch screws...

(1) After I'm done, I would like the currently switched receptacle to be:

(1C) Either way is fine.

(2) There is an accessible attic above this room.
(2A) True.

(3) It would be easier to fish a new cable:
(3A) From the ceiling to the switch.

(4) The switch and the receptacle:
(4A) Neither is on an exterior wall.

(5) The switch currently controls:
(5A) Only the top half of the duplex receptacle.

(6) There is a white wire connected to one of the screws on the switch.
(6A) True.

(7) The lever of the switch:
(7B) Is blank.

Thanks for the help!

So what is your question?

Fish a wire to the switch. Rewire at the receptacle so that it is no longer switch and so that the cable to the switch carries full power.

Wire at the switch to switch the hot wire.

Connect the wires to the light.

Done.

Sorry, my question is, can I just run the new wire from the light to the switch and connect the white to the white and black to the black on the switch screws? I don't care if the switch still turns the outlet on and off.