The best thing to do would be take all the media from your current, cycled filter and add that to the new one, then top it up with new media to fill the rest of the space (assuming the new filter is larger).
You could run both filters in tandem for a while (at least six weeks) but you may not get any bacteria growing in the new filter as the bacteria don't move around, and presumably your current filter is dealing with all the ammonia, so there wouldn't be any extra to feed a new colony.
You might find that when you remove the old filter, the new one won't have sufficient bacteria to deal with the bioload, so moving the media is the best way to do it.