If you wanted to just get the move to the bigger tank over with, You could move everything to the new tank including that smaller cycled filter and run both filters in parallel for a month, the colony on the smaller old filter will do the work,for the stocking you had, while the new filter will build a colony with a head start off the older colony that's larger, then remove the old filter in a couple weeks to a month, basically the same thing as starting the new filter on the smaller tank really.
when it's done you can move the small filter back to the small tank and use it for
QT or start another type of small fish or shrimp or something you want to do.
If I were to be doing this, it's what I would do, move everything from the 3g to the 6g ansd run the two filters in parallel and clean up the 3g to the bare tank,
buy what I want for the 3g for the new set up while waitng for that new filter to establish a colony off the old filter, and when that's done, set up the 3g new and put the 3gs old filter back on it and both tanks are cycled and good to go.
the one thing you lose switching tanks though is biofilm that takes time to build up. with shrimp the biofilm is quite helpful so they can graze when they want to, but not much you can do about it when you are switching tanks.