How to recover an overwritten SQL Server database
As far as SQL native backups are concerned, you can only restore to the last good backup that you have which is 4 days ago.  So, unfortunately, from a SQL server backup perspective you may have lost 4 days worth of data.

That being said, if this is a VM (HyperV, VMWare, etc) and there are snapshots, you may be in luck and your systems guys may be able to get the information back.