I would say you can move the fish into the new tank now.
I'd use the same filter, for starters, while you run the new filter in addition on the new tank.. Clean the old filter, gently, just before you switch tanks.. the same day or the night before at most. Keep the rinsings at tank temps 'til you use them. Pour the rinsings from the old filter into the new one. That will jump start the new filter on it's BB growth, and within a couple of weeks you can take the old filter off entirely.
Maybe replace it onto the old tank to use as a hospital tank ? Though you'd have to do something to keep the BB fed in that case. A fish or some snails perhaps. Having a QT is always a good thing. You can also have a sponge filter in the new tank, which you can switch to the QT if you need to use it, and not worry about keeping the second filter fed.
Or run both of them on the new tank so you always have a cycled filter to use on QT if needed.. very handy and overfiltration is not really possible.
