IPhone – Most recent backup not available when attempting to restore iPhone

you need to reset the phone to factory settings so that iTunes think's it's a new phone

it will then ask you if you want to restore from a previous backup and you can restore from any of them that are still on the system.

I found how to make iTunes restore only contacts, while preserving the rest intact:

1. run iTunes before connecting iPhone
2. disable auto-syncing in preferences: Edit > Preferences > Devices > Prevent iPods, iPhones, and iPads from syncing automatically should be switched On
3. connect iPhone to PC. In iTunes:
4. on the Sidebar, select your iPhone (press Ctrl-S if Sidebar is hidden)
5. on the top central menu (Summary | Info | Apps | Music | ...), click Info
6. scroll the page to bottom, there's Advanced section
7. switch on: Replace information on this phone > Contacts
8. click Apply in the bottom right of iTunes window (or right click on iPhone and choose Sync)

These steps wipe the local contact store on iPhone (leaving cloud contact sync alone, if applicable) and injects all the contacts stored in the PC-side data source that iTunes has selected (from Windows Contacts in my case; Outlook on Windows / Contacts on Mac in other setups) and pushes those contacts to the device. It does not load contacts from an iOS backup.