please help!

My son is 11 days old and since yesterday he's not been himself, a heatwave occurred two days ago and his temp went to 38.2 lastnight so I sponged him down with like warm water and it came back down. Today he is pale and very hot to touch but no temp, I took him to the doctors and they send us to hospital to be checked over. They watched him for four hours and discharged us without any advice at all. So how can I keep my baby cool? He's either in a nappy or short sleeved vest and we've got the fan on in the room for him so it's cool but I don't want him getting hot again because he will have to have bloods taken and I don't want that 😞