Transition from Moses basket to next to me crib

I’ve never swaddled my baby but he’s been in the Moses basket from the day we got him home as he wouldn’t settle in the next to me as it was too big and he was tiny but the past few nights I feel so uneasy with how big he is now. I wake up and his heads pushed up right at the top so I’ve taken him out and put him into his next to me crib but when he startles himself he wakes up a good few times and needs resettling. Is there anything I can do to help him stay asleep once he’s been put down or is this just a phase and he’ll get used to it eventually? On another note 😅 is it ok that he’s nearly 4 months old and still startles?