Lifting Corners of the Mouth by Mid-face Lift?

I don't think the long term effects of the cheeklift will be what you'd like as fart as the lifting of the corners of the mouth. I don't like the scars one would get with an incision at the corner of the mouth. You would do best with radiesse for this problem as the lesser of evils. 

A downturned smile or permanent frown can be improved with a combination approach, including neuromodulators (such as Botox, Dysport, and Jeuveau), injectable fillers, and radiofrequency skin tightening treatments such as Thermage or Venus Legacy. I recommend getting a formal evaluation with an experienced cosmetic dermatologist to develop a treatment plan that works for you. Best, Dr. Emer.

Hello tj in txA facelift will:
smooth out major folds and wrinkles,tighten the tissues beneath your skin,remove sagging and excess skin.A facelift can also reduce excess skin around the neck and jawline

A non surgical option for the mid face lift is Ultherapy, which can tighten the SMAS layer of the face. Raffy Karamanoukian MD FACS

I would not use the mid-face lift alone to achieve elevation of the corners of the mouth. It is more likely to restore fullness to the cheeks.
This can be performed using a variety of incisions from inside the mouth, to under the eye or through the face lift approach. It is not uncommon to combine fat injections to enhance the cheeks. Commonly overcorrection is required in order to compensate for long term settling. Initially, this may give you a very full cheek appearance.
I do perform corner lifts but usually on individuals over the age of 60 who have well established creases in which to conceal the incisions. I would also be concerned in a male that this could produce visible scars with possible loss of hair (cicatricial alopecia) which could make the incisions more visible.
Other non-surgical options which give the impression of a corner lift include the use of fillers to the labiomandibular lines.