Thanks, it would make accessing the backed-up saves easier, without polluting the active directory with so many files.

I move them manually but then the number sequence starts over — To then move the BackedUp files to the rest, it becomes a little more complicated:
1) Rename the files to be moved.
2) Move the files.
3) Sort by Date.
4) Apply a Rename/Mask with [C]ounter in TotalCommander’s MRT (multi-rename tool).