The LSL is usually a weak procedure for the midface. It does nothing in my opinion to permanently redistribute the deeper tissues that have sagged. It does nothing for the area below the eyes.
Depending on what arrangement you had with your doctor initially, he may or may not do additional surgery to help with your problem.
The lower eyelid / midfacial junction area is usually separate from the facelift, and is accomplished through an incision below the eye, in the mouth, or in the temple, whichever procedure is selected.
If you are that unhappy with your LSL, are you sure you want the same surgeon to perform a midface lift on you, when a midface lift is technically a much more demanding operation than an LSL?
Consultations with several board certified plastic surgeons with special interest in the midface may help you to sort out these answers.