Debian / Bullseye, should work on others
On the Old Server
systemctl disable jellyfin.service
systemctl stop jellyfin.service
On the New Server
Install Jellyfin on the new server
systemctl disable jellyfin.service
systemctl stop jellyfin.service
Copy the contents of /var/lib/jellyfin (Example)
mv /var/lib/jellyfin /var/lib/jellyfin.old
rsync -auv oldserver:/var/lib/jellyfin /var/lib/
chown -R jellyfin:jellyfin /var/lib/jellyfin
Copy the contents of /var/cache/jellyfin (Example)
rsync -auv –delete oldserver:/var/cache/jellyfin/ /var/cache/jellyfin/
chown -R jellyfin:jellyfin /var/cache/jellyfin
I would reboot
systemctl start jellyfin.service
Verify everything is working
systemctl enable jellyfin.service
Reboot to verify auto-start