I backed my iPhone 6 up to my new Macbook Air, but the backup did not save.

I backed up my phone and then gave my new computer the authorization to sync my phone to iTunes. After that, it told me that in order for this phone to sync, they had to replace the material on my phone from my other computer with what was on my Air. It then deleted all of my songs. In an attempt to restore my phone, I tried to use the recent back up-but it was no longer there. I had to use a backup from June 9th on the computer to restore my phone, but now I have nothing from after June 9th on my phone.

Long story short, is there a way I can recover that lost back up?

Thank You!