I packed my son who is also due in June two onesies, one tank and one short sleeve and shorts to go with both, he may not wear them, but then again, he may. I also packed size 0-3 of the same thing basically just in case he's a chunker.
june is worm so not long sleeves maybe just a couple cute onsies and a small blanket for the ride home. pacifiers, and some diapers, wipes depending if the hospital will provide them or not.
Pack and outfit for yourself and one for baby can be just a onsie cause he's due in summer time also pack bottles passie and a blanket just in case of rain