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'm 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.

Firstly, if you do not have any new data on your iphone, then you can use iTunes backup to recover your old data to your iphone. (note, this backup will remove the current data of your new iphone.)
Secondly, iTunes can only recover some data, like music, videos. But contacts, call history, messages, notes, etc.can't be recover by iTunes.
Thirdly, you can use iCloud backup for several times to back up your old data, then recover them for several tiems. But that's time-consuming.
Last one is to use a tool to transfer all of them for one time. That's safe and easy. My friend has used FoneTrans to remove them. It has the free trial, you can ahve a try just for future's use. Hope it can help you.

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