deleted startup disc

At some time and with some filesystems, you could unformat a partition. MS DOS 6 or 7 had an unformat command. There are still some unformat utilities around. I have no idea how well they work.

Was your old startup drive written over? What does it look like now?

How badly do you want your data back? You could give drivesavers.com a call.

"TestDisk is a free data recovery utility. It was primarily designed to help recover lost data storage partitions and/or make non-booting disks bootable again when these symptoms are caused by faulty software, certain types of viruses or human error (such as accidentally erasing a partition table)."

http://en.wikipedia.org/wiki/TestDisk

Wondershare Data Recovery for Mac

"Recover lost photos, video, documents, email and archive files from any Mac-based hard drive or external hard drive, iPod, USB drive, SD card, digital camera, mobile phone, and MP3/MP4 player, due to accidental deleting, formatting, virus infection, improper operation, unexpected power failure, and other unknown reasons. Also, iPhone recovery allows you to recover lost photos, videos, SMS, contacts, note, calendar and call history from iTunes backup file. With the free trial, you can preview all your recovered images, videos, music, documents and archive files prior purchasing this Mac Data Recovery."

http://software.bigbigsoft.com/data-recovery-mac

If you really need your data back, I suggest drivesavers.com. Give they a call and they will give you a quote. Or they may have you send the drive and for a fee, they will provide a quote for recovery.

You could buy an external drive. Install macOS on the external drive. Install one of the above mentioned utilities and give it a try.

Robert

From the backup you made on a separate drive before you reinstalled the OS. If you haven't got one, you're stuffed, I fear.