iCloud backup

I just got a new iPhone and when I signed into my Apple account I got everything from my old phone like normal. But I didn’t have enough iCloud space on my old phone to back the rest of my data up to the new phone. So I bought more space today but I cannot get my old phone to backup so I can put the rest on my new phone. I’ve tried logging out of my Apple ID but that’s not working either. I need help!

Then after I get everything backed up on my old phone, how do I get it on my new phone? Just log out of my account and log back in?

please help! Thank you.