As you know for a domain to be hosted somewhere it has to have nameservers attached and in normal circumstances and with most hosts you cannot change the nameservers for a subdomain and they will usually have to stay with the current host.
However it is extremely easy to forward a subdomain to another server as long as you can create “A” DNS records with your domain/web host;
1. Set up the domain hosting as normal on the destination host in anticipation of the move.
2. Get the IP address of the destination server (even if this is a shared server) for example 18.104.22.168.
3. In your origin server’s control panel software go into the DNS settings for the parent domain for example domain.com.
4. Find the option to create an “A” record.
5. Input the subdomain for example other.domain.com.
6. There should be another field for IP address and this field will only work with an IP, so input the destination server’s IP.
7. Click save/add/finish, whatever your host has to complete.
8. You should now have an A record pointing at your desintation server IP.
9. Wait up to 48 hours.
You should now be able to browse to the subdomain you created and be directed straight to the domain space you set up on the destination host, as an end user you should see no difference in URLs from the destination IP.
NOTE: This definitley works with shared hosts (where you do not have a dedicated IP) I have tried and tested it.