Hi Kiera
I’m pretty sure they do pay private rent, as when I had to stop working for a while, it covered my rent, and I was in a private let.
That may differ from council to council, and I couldn’t be sure as to whether it makes any difference as to being in the property all ready, or looking for one. But I can’t see it personally.
It would help to ring the relevant council that covers your parents area, and the job centre and tell them your sitiation, and your hopeful plans and they should point you in the direction.
As far as I’m aware, I think the only drawback of getting a private let is, it would put you back getting a council house, but as you’d all ready have one, it wouldn’t be so much of an issue.
Hope that helps a bit.
Adam