The current theories of facial aging provide that facial fat melts away in some areas and accumulates in others. The corners of the mouth exemplify a zone where soft tissue "melts" away, leaving a trench, starting from the corner of the mouths and extending inferiorly towards the jaw; the so-called "marionette's lines".
A lower facelift assists in repositioning the mandibular jowls and redefining the neck and jaw. If a surgeon exerts adequate pull on the overlying skin and muscle, much as you do, when simulating the effect of the face-lift, partial improvement of the marionette's lines will result. However, if, as you observe, there is still a line, then a dermal filler, Hyaluronic acid, Radiesse or Evolence will provide structural support and improve the final result.
Prior to proposing any surgery, I ask patients to identify features they dislike and then we discuss "solutions" as well as recovery time, longevity and possible complications. Hope this helps. Good luck.