can I go home the day I have my baby? Or can my toddler stay overnight at the hospital?

Worried sick about leaving my 3.5 year old overnight the day our 2nd child is born. My husband refuses to leave my side and he doesnt think its a good idea for me to go home the same day I give birth. If we get a private room, will he be allowed to stay overnight? He sleeps alone in his bed at night but always comes into our bedroom and ends up in my arms til he wakes up. No I dont want to change the way he is because its not like he'll be that way forever. He's only little for so long. I'll take all the hugs and kisses I could get before he wants nothing to do with his parents lol. My family can watch him but I know he'll cry in the middle of the night. Husband's parents live overseas. Husband keeps saying he'll pay extra for a private room so our son can stay with us. Has anyone ever done that?