後炭 is a bit tricky here, and the books are not sufficiently clear about it. The main issue is that there are several orientations from which the is handled and shifted. Grosso modo there are two ways. I think:

In general, I feel 🅱️ feels nicer. It also ends up to be the way where you can pick up the 鐶 from where you put them down.

Then it's business as usual: 初掃・直す・香 and 灰・中掃 (abbreviated!)・炭・後掃.

Come in with the 水次, sit to face the , replenish and wet, then leave. Remember that with the 片口, the lid stays on the floor until you've finished wetting the .