Preposition before "halfway"

Other than this discussion on the Free Dictionary forum

• I always notice that "halfway" comes without a preposition. Thus, I am asking whether or not I can say "At halfway" ...
• Halfway indeed comes without a preposition ...

and the existance of few reasonable examples on the internet, it's hard to prove the negative

'We usually part company with each other at halfway between our homes' is unidiomatic. And indeed, 'There is a checkpoint at halfway between Rome and Naples' sounds far more natural.

But I'd never use the given sentence, preferring

'We usually part company with each other at the halfway point between our homes'

or better

'We usually part company halfway between our homes'.

I've seen this latter usage referred to as that of a 'prepositional phrase modifier', other examples being 'just' and 'straight' in 'The cat is just behind the door' and 'She pitched the ball straight into the hole'.