It depends how many domains you are allowed to host with your Siteground package.
If it is more than 1, you can simply add 'Domain 1' to your existing Siteground account and then change the name servers for 'Domain 1' with your domain registrar to point to the Siteground name servers.
Backup all the data for 'Domain 1' with hosting 1 and restore to hosting 2 (Siteground).
You might want to change the DNS TTL setting on 'Domain 1' to something like 600 (seconds) 24-48 hours before your perform the migration, to help ensure a quick change over when you update the name servers.
This way, both domains will be running from the same hosting account with Siteground.
After you have done that you should setup email forwarders from any Domain 1 email accounts to Domain 2 email accounts, to ensure you don't miss any emails while you fade the old Domain 1 out of commission. If the email addresses are setup the same on Domain 1 and Domain 2 (e.g. email@example.com
), you can just setup a Email Domain Forward from Domain 1 to Domain 2.
Hope this helps.