Regrowing moss in yard

Hello! Looking for advice on regrowing moss in a yard in North NJ. We bought a fixer upper on a little lake 1.5 years ago. We just redid the lake retaining wall and the yard by the wall was destroyed in the process. It’s all muddy dirt right now. We used to have a lovely (I think natural) light layer of moss back there. There is tree cover so it’s usually shady. We have no idea what we are doing but told the contractor not to put any grass seed down.

1. Will the moss grow back naturally?
2. If not, would a moss slurry using moss from other parts of the yard work?
3. Given that it’s October and starting to get cold what can we do now vs wait for the spring?

We do not have much budget to do a big transplanting project right now.
Thanks in advance