Can fat grafting (or fillers) under my eyes slightly lift their outer corners?

Fillers, fat, lasers, microneedling/prp helps this area.  I suggest seeing a cosmetic dermatologist with experience.  Best, Dr. Emer.

Without photos it is hard to recommend a procedure.  If you have some hollowing, then filler should help.  Filler placed toward the lateral part of the lid can help to support the lid, and may even raise it up some.  Before doing fat try some filler to see if you can achieve the look you want with extra volume.   

Fat grafting also called fat transfer can certainly be used to improve hollowness under eyes. Results look and feel completely natural and last for many years. A careful exam would need to be done to determine if the added volume under the eye would help support the lower lid and corner or not.