Contacts Disappear After Restoring from iTunes Backup

I was recently given a new phone by Apple and I am trying to figure out why my contacts are not restoring from the backup properly. I backed up my phone to iTunes last week and am now running the restore from backup process. As soon as my phone turns on after the process is complete, I notice for a few minutes that my contacts remain on the new phone but quickly disappear. When I sign in to my iCloud account online, there are no contacts displayed in the "Contacts" section. I am assuming I did not have my contacts backed up to an iCloud account, but I figured I wouldn't have any issues given they were recently backed up to iTunes. What steps can I take to recover my contacts or back them up to an iCloud account? I am limited in options because I seem to only have access to the contacts for few minutes after I restore from the iTunes backup. I have tried many of the options outlined on this forum and haven't had any luck. Any advice would be much appreciated. Thank you.