So long as you keep a backup before trying this, this is how I’ve dealt with that before:

Assuming you’ve initialised the new card in your 3DS, your SD card now looks like that screenshot, correct? If so, go into the first of those folders in your screenshot. There should be two folders – copy (or move, if you’ve a backup) the contents of the older folder into the newer one, saying yes if it asks you to merge.

Then go into the second of the folders in the screenshot, and, if there are two folders in there, copy the older into the newer again. Finally do the same for the third folder.

When this happened to me (when I was doing it for someone else), only one of the three folders actually had two folders in it. One of them had one (so I left it alone) and the third folder was totally empty.