No, you would put this file and compressed archive on FAT partition (first one). But you should be careful with a script not to break boot process. And archive must be extracted only on second boot after storage resizing (when file .please_resize_me doesn't exist anymore).
Ah, just forget my suggestion. It is to complicated
Maybe it doesn't even work as I though. The file should be mount-storage.sh and storage partition must be manually mounted from this script.