Did you, by any chance, use iCloud for Photos? I ask because if the photos were not in the backup file then the Photos may be on iCloud. If you have photos in iCloud then all it would take is to log in with the same Apple ID on the new iPhone and the photos should begin migrating / synching from the old to the new iPhone.
Of course, if you do not use iCloud for photos then the information above will not work. At that point, if the photos are not in the backup file that you had created and they are not in iCloud then you have lost your photos permanently. I am hoping, for you, that they exist on iCloud.
Axel F.