Have you thought of trying wool or fleece? Wool is the more reliable under clothing and if baby is in one position a lot (which newborns are!) and of course they are super soft and super stretchy so much more flexible than any PUL wrap available.
You can use wool at any time. Just make sure you dont put cloithes that are snug arround the bum otherwise you can get compression wicking. We use wool soakers under poppered vests and sleepsuits, and we also use longies instead of trousers.