If you have k predictors, X_1, X_2, ...., X_k that are mutually orthogonal and you add X_{k + 1} that is also orthogonal. My intuition says (i.e. I haven't worked it out myself) that this is sufficient and easy to show.

You may be able to get away with a less stringent condition that is the same in spirit (i.e. rooted in orthogonality) but the problem seems more difficult then.