Transferring data to new iphone using itunes - but still using old phone?

I have an iphone 5 and now need to transfer everything over to an iphone 5s

I tried icloud but it said I didn't have enough storage to complete the icloud backup.

I then used itunes to backup everything on my old phone,

However, my EE number is being transferred and will take up to 48hrs so I am unsure as to when my old phone will become inactive.

Can I backup my new iphone using the itunes backup, but still have access to the messages/photos on my old phone?

I will be at work tomorrow when the phone number is being transferred so ideally I would like a phone that I can use.