Assuming that the red is the switched wire, the one that turns the half of the outlet on and off, yes. Splice your new black wire to that, white to white and ground to ground, and take it to the new lighting. If you want your outlet to remain on all the time, splice the red wire to the outlet to the black coming from the switch location instead of including it with the new black wire and the red from the switch.
Unless you have already checked, though, be aware that the wire from the switch may not ever go to the attic - it may stay in the wall. If it does, best bet is probably to take the new wire to the switched outlet, performing the same splices there.