If my partner ever left soon after the birth of any of our babies, whether it be the first or 7th, he'd be told to not bother returning....

I refuse for anyone to have my babies for the first few hours except me and Dh. After babe has fed, is wrapped and asleep, Dh has bubba while I shower and get cleaned up, and then helps me to my room. This can take up to 3 hrs, if babe feeds for a while, and you need to lie down and rest a little after birth.
I tend to bleed quite heavily for a few hours, so he gets anything I need, and passes bub to me, so I can stay lying down. Regardless of visiting hrs, he has never been kicked out...

It can feel quite lonely to be suddenly left alone with babe too... you may just need him for company too....