Cambridge Grammar of the English Language has this portion regarding backshift on page 154:
- Backsift not confined to indirect reported speech
We introduced a concept of a backshifted preterite with an example involving indirect reported speech, . It is important to emphasise, however, that the backshifted preterite is used much more generally than in reported speech. Compare:
 i This meant that Jill had too many commitments.
ii That Jill had too many commitments was undeniable.
(Bold emphasis mine.)
The that-clause of the former example sentence functions as a complement to the verb "meant", whereas that of the latter the subject of the sentence. These are clearly outside the scope of reported speech. So CGEL does say the backshifted preterite (= past tense) can go beyond reported speech but does not clearly say how far it can go. In fact, CGEL fails to specify whether the backshifted preterite is applicable to other types of subordinate clauses than those mentioned above.
EXAMPLES FOR QUESTION
So I have laid out three example sentences as follows. (Boldfaced are the preterites that CGEL might--but fails to specifically--recognize as "backshift".)
(1) I assumed you were the type who kept your promises.
This one was borrowed from an earlier question. The clause "you were the type" is a complement to the verb "meant", where "were" is recognized by CGEL as "backshift"; the clause "who kept your promises" is a relative clause, where "kept" is not recognized by CGEL as "backshift".
(2) She was surprised that he still loved her.
The clause "he still loved her" is a complement to the adjective "surprised", where "loved" is not recognized by CGEL as "backshift".
The way people experience their iPhone has always started with the display.
(3) So we wouldn't introduce a larger display until we could make one that was great.
This is part of what an Apple's senior VP has said in a promotional video of iPhone 6. The clause "we could make one" is the complement to "until", and the clause "that was great" is a relative clause, where neither "could" nor "was" are recognized by CGEL as "backshift".
And the big question is whether any of the boldfaced verbs can possibly be subsumed under "the backshifted preterite" as defined in CGEL?
CGEL at page 153 presents "conditions for backshift" in , along with relating examples in  and .
A backshifted preterite can occur when either of the following conditions obtains:
 i. The tense of the matrix clause is past.
ii. The time of the matrix clause situation is past.
The question of what types of subordinate clauses are covered in  hinges on the meaning of the term "backshifted preterite" at this point in CGEL (at page 153). Indeed, CGEL has this term defined at the bottom of page 151 leading to the top of page 152, as follows:
...the term backshifted preterite is intended to suggest this change from an original present to a preterite. We retain the traditional terminology for its mnemonic value, but emphasise that backshift is not conceived of here as a syntactic process: we are not proposing that had is syntactically derived by changing the present tense of an underlying have into a preterite. The issue to be considered is simply what the preterite means in this construction.
(Only the latter bold emphasis mine.)
The above-mentioned "this construction" at page 152 refers to indirect reported speech, and there is no change in the meaning of the term "backshifted preterite" between  of page 151 and  of page 153, in terms of in what construction it is being discussed. Therefore, it is clear that the meaning of the term "backshifted preterite" in  is limited to indirect reported speech, and that the types of subordinate clauses covered in  are at best limited to content clauses functioning as a complement to a verb of the matrix clause.
In fact, this is corroborated by the first boldfaced sentence (on page 154), which is shown at the top of this question. If the conditions set forth in  of page 153 were to cover other types of subordinate clauses than content clauses functioning as a complement to a verb of the matrix clause, CGEL would not have inserted, only after , the section of "backshift not confined to indirect reported speech". And even then, this section, as presented at the top of this answer, doesn't really go far beyond content clauses functioning as a complement to a verb of the matrix clause. The only expansion shown in this section is that the content clause may function as the subject of a verb of the matrix clause, as in [21ii] (That Jill had too many commitments was undeniable.).