You seem to be asking whether one clause can be a constituent of another clause. Yes, it can. That's the reason that terms like "main clause" and "matrix clause" exist.
As I parse your example, you have one sentence which contains one matrix clause which contains one subordinate clause. The subordinate clause is an adverbial clause, and it acts as an adjunct of the matrix clause's verb. The subordinate clause is part of the predicate of the matrix clause.
Not all subordinate clauses are part of a matrix clause's predicate. That they have another use is easy to demonstrate.
Consider the sentence "That they have another use is easy to demonstrate." The subject of this sentence is a subordinate clause, which has its own subject. We can divide the matrix clause into the subject "That they have another use" and the predicate "is easy to demonstrate." We can divide the subordinate clause into the subordinating conjunction "that", the subject "they" and the predicate "have another use."
I do have a small problem with the way you phrased the following question:
- Is it possible for this subordinate, adverb clause to be part of the entire sentence's complete predicate?
That's a common phrasing and it's not completely wrong, but it's not completely clear, either. Your example sentence has one independent clause. That one clause has one subject and one predicate. However, compound subjects, compound predicate and compound sentences do exist. It won't always be true that a given sentence has only one main predicate.
"The predicate of the sentence" is shorthand for "The predicate of the (one) main clause of the sentence". In your example sentence, there's no such thing as the one complete predicate of the sentence. The sentence contains two predicates, one inside another. There does happen to be only one main clause and only one predicate of that main clause, but two predicates in it.
When I rephrase your question without the shorthand, it practically answers itself.
- Is it possible for this adverbial subordinate clause to be part of the predicate of the matrix clause?
If it's subordinate, then it is contained in a matrix clause. If it's adverbial, then it can modify a verb. If it does modify the verb, then it's an adjunct of the verb and a constituent of the predicate. That is to say, if the words in my rephrasing of the question are the right words to use, then it must be possible.