My mother is a retired midwife and she would advise "don't give them a pillow until they ask for it." Though literally an old wives tale :). Perhaps give the health visitor at your clinic a ring?
Hello, would I be able to join the club? I live off Lordship Lane and my daughter Violet was born in April but only recently discovered the forum. I'll keep an eye on the forum and see if I can make it out to a meet up. Rachael