If you use iTunes backups, there is only one backup (the latest) maintained per device at any point in time. So in this case the new backup overwrote the older backup, and it would be possible to recover only if a copy of Library/Application Support/Mobile Sync/Backup/ was made before the clean install (through Time Machine or any other solution), restoring that folder on the system and doing a restore from iTunes.
You can see the device backups (with the time stamps) available in iTunes by going to iTunes → Preferences and then navigating to the Devices tab.
Some recommendations and points to note to avoid such a situation and have something to fall back to:
You can have multiple backups in iCloud for a device and choose which backup to restore from (assuming there are no device/iOS compatibility issues).
Use both iCloud and iTunes backups together (once iCloud backup is turned on, automatic backups in iTunes would be turned off; so a manual backup would have to be triggered in iTunes).
When using iTunes backup, trigger a manual backup in iTunes followed by a manual backup in Time Machine before major changes to a device, like re-installing iOS or jailbreaking or restoring iPhone from previous backup.
You can learn more details from here about:
How to restore iPhone from backup
