Do I need a new Pokemon Bank subscription after doing a system transfer

I just did a system transfer from my 2 year old 3DS to a new 3DS XL. With the current version of the 3DS software, after the transfer is done, the new 3DS wants to use the old 3DS's SD Card. So going through the standard System Transfer process would do exactly what you want!

In my case, I wanted to use the new 3DS XL's SD card since it was bigger than the old 3DS's one. Nintendo has a pretty easy process for this: http://en-americas-support.nintendo.com/app/answers/detail/a_id/309. It's essentially copying the contents of the SD card from old one to the new one via your computer's SD card reader.

Unfortunately, as you have noted, without the original system to transfer to, you are likely without hope. 3DS titles are tied to console, not to any ID (Nintendo Network ID or otherwise), so without the console, you can't do a whole bunch. If you had the console, you'd be able to do things differently, but as it stands, there isn't much you can do.

There are stories online where people say you can contact Nintendo to get your stuff back, but since Nintendo themselves seem pretty hush on the matter (their website states that you can only do things through transfer), I don't know how lucky you will be. Club Nintendo keeps a sort of record, in a way, so they might be able to help that way.

I would also keep in mind that stores usually wipe any used consoles they receive, so even if you can get to the store, it is not necessarily true that you can even get your stuff back, even if they have your particular console still, as doing a format wipe of the SD card and the system means that your information will also be gone. This prevents the next owner from being able to essentially get free games by downloading what you had already purchased.