Complete contact list did not transfer

Contacts are associated with email accounts, and frequently contacts are spread across multiple accounts. So first, did you set up all of the email accounts that your previous phone had installed? And did you enable Contacts in the setup of those accounts?

Next, in Contacts, does it say Groups in the upper left? Tap on it, and make sure all groups are checked.

If all of your contacts were synced to iCloud, here is another way to recover→Restore contacts, calendars, bookmarks, and more using iCloud.com - Apple Support

Transfer from what? How were you managing these contacts? Is it possible that you had contacts from more than one source?