How do I find a locking doorknob for an existing door with a non-standard backset?

I’m trying to replace a lockless interior door’s knob with one with a lock (not a "privacy lock", but an exterior-door-type key lock). The existing hole and knob has a non-standard backset of about 2-5/8". I’ve found plenty of new knobs that are adjustable between the standard sizes of exactly 2-3/8" and 2-3/4", but nothing that can be adjusted to something between those sizes.

Is there any way to find a knob that will fit without re-drilling or replacing the door?