On established tanks I have put a new filter on and moved the old filter to a different tank, and did not use old media or anything - brand new media and filter - and did not have a bit of trouble with a bio-overload. The tank walls and decorations, etc. has plenty of bacteria, IME
, to handle a completely new filter being put on and the old taken off.
If you have any concern, either run them both together for a couple of weeks or put the old media in the new filter for a while.