Can the corners of my mouth be lifted during a lip lift?

I’m planning on getting a lip lift in the coming month, and was wondering if I can have my corners lifted at the same time, and whether or not it’s considered a different procedure. I’m just 29 years old (I look even younger), but I’ve always had this minor downturned mouth. In my head it seems like the lip lift itself could just focus more on lifting the outside of my mouth instead of the middle. To avoid making a “cat mouth”. What do you think? Thanks!