A predicate needs its argument to complete the meaning. An object is one as such (direct argument); the preposition phrase is another (oblique argument) but very much essential to the meaning of the sentence. The preposition is the argument marker.
Not going into pedantic details, we can say this much that in both sets of examples
WAY is used all along as a noun. We don't find fault with the first set of examples —
IN ANOTHER WAY — is an
Adverb Prepositional Phrase (preposition in the beginning, its object at the end and a modifier in between; in - another - way). It fulfills oblique argument in the first set of three.
What about the rest?
WAY is not an adverb here.
WAY or for that matter
ANOTHER WAY is an
adverbial objective (they create an object like illusion though not acted upon) or
adverbial noun i.e., nouns used as adverb:
He works mornings and nights.
Mornings and nights occupy the position earmarked for an object. But a word of caution: adverbial nouns modify verbs and adjectives and they are not used to describe manners.
However, in the first two of the remaining set of three
another way functions as adverbial objective – localising the functionality of the verbs. But in the last sentence we want
another way to function as an adverbial objective assigning the role to describe manners which is contrary to its nature. The last sentence flouts basic rules of semantics and is wrong as such:
He is used to building roads another way.
In this example sentence the participle object phrase
building roads can be classified as an object of another object
roads usurping (leaving no room for) adverbial objective (yet another noun —
another way). So we have:
Without a relationship word (preposition)
another way can not be knit in the sentence coherently or logically. However, I have a sort of inner prompting that says that if
roads be replaced by any pronoun (it/this/one) than
another way may be induced in the sentence without a preposition but the rule underlying is beyond my knowledge.