When we bring our baby home it will be September so we will have baby in just a long sleeve and pants, no blanket unless its chilly or rainy.
I would be careful with a coat in June as it usually gets hot and your baby can overheat. Personally I would get a thin and a thick blanket. Depending on the weather use the right blanket to wrap around your baby's clothes. Play it by ear:)
For baby just a going home outfit: long sleeve onsie and pants or a sleeper So they wont need a jacket or sweater. If you go with a short sleeves outfit id go with a thin sweater seeing as june over here its hot idk where you are and it may be cold..