Are you using iCloud to sync your contacts? If yes, try to disable syncing for contacts in Settings/iCloud, tap on "yes" if asked if you want to keep the contact info on the phone. Then try again to search for contacts like the way you did.
If this does not work, and to make sure that this is not software related, set it back to factory settings, without using any backup data afterwards. Set up the rest of the personal settings manually add some contact info and test the function again.
If the phone works as before the update, restore from the latest backup. If the problem comes back, the backup data contained corrupt data, causing your issues.
If it still does not work, the phone has to be checked by Apple or an Authorized Apple Service Provider:
Use iTunes to restore your iOS device to factory settings - Apple Support
Find an Apple Authorized Service Provider
iPhone - Contact Support - Apple Support