As an example, if your domain is example.org and you want to point the subdomain new to a server with a different host provider you would:

1.) In the DO control panel, go to Networking->DNS and select your domain
2.) Create a new A record. For the name you would put new (the new subdomain without the rest of the domain) and in the other field you would enter the IP address of the server with your other hosting provider.

Once this is done www.example.org and example.org should be unchanged but new.example.org will now load the content from the other host.