Transitioning out of Swaddle

We have begun transitioning our son out of his swaddle. He isn't doing do so well with it. He will start off great, but as the night progresses he wakes up 2-3 times. Before he would sleep all night 9-5 and then sleep 2-4 more hours after that. I dress him in a cotton sleeper then put him in his sleep sack. He is still in his bassinet, which he is about to outgrow. Hoping my husband will help me rearrange our room soon so we can have room for his crib. I'm not sure he is having trouble adjusting because he loves the snuggness or because his little hands get cold. Any advice would be appreciated.