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.

You can reveal more lateral lip by performing and Extended Lip Lift or by performing and modified upper lip lift with lateral release.   These can both provide improvements in lateral show but extended lifts tend to scar more given the increase in lateral tension.  If a more substantial amount is needed then a corner Lip lift can be performed as well at the same or at a later time.   I would be much easier to advise you with some photos

Bullhorn  lip lift aims at reshaping central part of upper lip only. Results in shortening philtrum & improving shape/ volume of the upper lip. Lip corners can be treated by corner lip lift to raise them & design more cheerful look. 
best regards, 
Piotr Osuch, MD, PhD