The claims in the Matrix Wiki's article on Residual Self Image is a bit difficult to follow:
Zion operatives who re-enter the Matrix appear as avatars with
appearances that can be totally different from their true appearance
within the real world.
"Appearance" here probably means clothing and hairstyle, not physiology. The former aspects of appearance can and do change. Clothing, it seems, can be loaded at will. Hairstyle seems to reflect the person's self-perception (like Neo's in the sequels).
A person jacking in could probably manage a false mustache but not much more. It seems their physiology is picked for them by the Matrix itself. The only exception I can think of is Bane. After the infection, he looked like Smith within the Matrix, but obviously, looked like himself in the real world.