iPhone backup data overwritten by mistake
With iTunes backups, there is only one backup (the latest) maintained per device at any point in time. Since you have overwritten the older backup with a new one from the fresh install, it's possible to recover only if you have a copy of the backup folder made before the install through the built-in Time Machine backup system on OS X (if you have a Mac) or any other solution.

Here are the iTunes backup folder locations on different operating systems:
OS X - ~/Library/Application Support/MobileSync/Backup/
Windows (XP, Vista and 7) -  %APPDATA%\Apple Computer\MobileSync\Backup\  

Since you do not have iCloud backups, you're completely out of luck if you didn't have backups of the iTunes backups. If you do back up your computer, simply restore the folder where iTunes stores iOS backups (after making sure you have a current backup for other devices if needed) and you can restore your iPhone from an older backup. 

For more information, see these answers to similar questions - they provide details as well as recommendations for the future:
Restoring an OLD iPhone backup
Recover iPad Safari bookmarks overwritten by iCloud