migrating / merging iphone contacts
You could restore the old backup to your iPhone 6 and sync those contacts with iCloud, then sync contacts on your new iPhone 7 to get those. (As fsb noted above)
Some third party backup reading tools allow you to export contacts. I'm a developer for Decipher Backup Browser - you can use Decipher Backup Browser to export the contacts from the backup as a VCF file, and then email/text/dropbox/etc that to yourself on your new iPhone to import the contacts. (When you open the attachment on the iPhone 7, one of the built-in VCF actions is to import to contacts.)
You could setup a gmail or similar account to sync contacts temporarily -- basically the same idea as syncing contacts with iCloud, but not iCloud if you're opposed to that.
I would not recommend it, but you could restore the backup to the iPhone 6, and then go contact-by-contact pressing Share to send the contact to your new iPhone.
Do you have a Mac? You may be able to restore to the iPhone 6, sync contacts on the 6 with iTunes, and then share multiple contacts to your iPhone from the macOS address book app? https://www.tekrevue.com/tip/share-contacts-mac/ (disclaimer: I didn't try doing this to get the step-by-step. It's just an idea.)