Get a sponge from your existing tanks filter.
Squeeze the goop out of it into the new tank.
Let the new filter suck up the goop, and the attached cycle bacteria
Now you have seeded the new tank, and it will be safe for at least a small amount of fish right away.
Taking a whole dirty filter sponge and stuffing it into the new filter works even better.
But you are on the right track. Once you have one cycled tank you dont need to cycle from scratch again. Just clone the cycle to the new tank by moving filter media around.
Ian