Give the background its own lighting job
Assign the key, background light and rim light different areas instead of letting every source illuminate everything.

The jobs of the three lights in the photograph
Read the left key on the face, the small lamp aimed at the background and the strip source near the shoulder in the photographic example above.
- 1Key → face
- 2Background light → wall
- 3Rim → shoulder
A pale wall can photograph dark
Recorded background tone depends on the light reaching the wall relative to the exposure chosen for the face. Separating subject and wall gives room to control these regions. Distance alone cannot make a wall black: windows, ceiling bounce and other lamps still matter.
A background light is not a second key
Aim a background source at the wall to create a pool or gradient, and prevent spill onto cheeks or the lens. Keep the face key from washing the whole wall. A grid limits spread; a flag blocks a particular route. Their purpose is control over assigned regions, not automatic darkness.
Leave an edge only where it helps
A little light on hair or shoulder can clarify a dark subject against a dark background. It need not outline the entire person in white. Check whether the rim source enters the camera directly and causes flare or lifted blacks. The photograph above illustrates those three distinct roles.