When using the RecoveryISO that support can mount for you, one option is to set up an SSH server in the recovery environment. To do this:
Mount your disk from the menu
Configure Networking from the menu
Select the option to set up ssh
The ssh server will be downloaded and installed and when it's complete you will be shown a temporary password. You can use this password with the root user to copy files over sftp/scp.
To do this from your new droplet and copy everything in /mnt to /root/backup on your new server (assuming the directory /root/backup already exists) you do do:
scp -r root@[old_droplet_ip]:/mnt/* /root/backup/.
This will copy the entirety of the /mnt directory over. In actual use you'll likely want to select just the files you need to move (like
/var/www/html) and do something like:
scp -r root@[old_droplet_ip]:/mnt/var/www/* /root/backup/.