Limiting modifiers such as only, even, almost, nearly, and just should appear in front of a verb only if they modify the verb: At first, I couldn’t even touch my toes, much less grasp them. If they limit the meaning of some other word in the sentence, they should be placed in front of that word.

Only limits the meaning of the when clause.

The adverb just limits the meaning of chemistry majors, not interview.

When the limiting modifier not is misplaced, the sentence usually suggests a meaning the writer did not intend.

The original sentence says that no black southerners were slaves. The revision makes the writer’s real meaning clear: Some (but not all) black southerners were slaves.