This is the exact way I do it:
I place a plate or a follower the diameter as close as possible to the mould’s inside diameter on top of the cheese.
Holding with both hands the “sandwich” I turn it upside down. Now the mould is on top which I gently pull up (like a hat).
Now I place a similar plate/follower on top of the cheese and turn it over again, now the first plate/follower is on top, which I remove now, and put the mould like a hat on top of the cheese and gently push the plate+cheese into place.
Now, still holding the plate/follower I turn the whole thing again and remove the plate/follower until the next flipping.
As the cheese dries and gets firm, the whole thing is easier and usually I menage without the plates.