You can 'touch' each file as it is put into the folder, which will set its modification time, and you can sort by that. You lose the actual modification time, though, which may not be ideal.

If your filesystem stores last access time (atime), you should be able to sort by that (possibly peeking at the file when you put it into the folder to change the atime). That's not an ideal solution either, since it's too easy to update the atime by accident, and many filesystems don't save it.