Will a bullhorn lip lift alone help with lifting the corner lip? Or is that addressed with a different procedure?

It is addressed with a different procedure, called a corner lip lift. The traditional lip lift and the Elelyft will only address the portion of the lip under the base of the nose. A corner lip lift may result in visible scarring as the incision is made along the vermillion border. Gary Linkov, MD
Lip Specialist / RealSelf Lip Lift Medical Reviewer
Manhattan

Corner lip lifts are typically performed by placing the incision at the corners of the upper lip at the lip line.
However, this can result in visible scarring.
So, an alternative is performing a Lip Lift, focusing more so on the lateral aspect in order to pull the corners up more than the central lip, in order to achieve your goal.
Keep in mind not all lip lifts are the same. A proper lateral release is important in being able to lift the corner up without a corner lip lift. And, you may still need a little bit of a corner lip lift. Photos would be helpful.

The technique I recommend is an extension of the bullhorn lip lift.  There is more tissue excised laterally than centrally to pull up the corners.Best Wishes,Gary Horndeski, M.D.