One option is to create something like archive.newsite.com.
Do all of this for your existing FTP user (so it's in your current FTP directory):
1) Create Fully Hosted Domain for newsite.com
2) Create Fully Hosted Domain for archive.newsite.com
3) FTP in and rename archive.newsite.com to save.newsite.com
4) Rename oldsite.com to archive.newsite.com
5) In the panel, change Hosting for oldsite.com to be a Redirect to archive.newsite.com (if you want the old site to continue to live in some form). Once registration expires, this will die, but archive.newsite.com will live on.
Make sure that oldsite doesn't have any links still pointing to oldsite.
Now you can fully set up newsite.com with links to archive.newsite.com/pic/coolpic1.jpg