Hey there Jac017,
Thanks for reaching out to Apple Support Communities. We understand that you're unable to restore your new iPhone from a backup you have due to a lack of available storage space. We're happy to provide some assistance.
Check the available storage space you have on your new iPhone by going to Settings > General > iPhone Storage. Afterwards, check to see the storage space that is currently used on your old iPhone by also going to Settings > General > iPhone Storage. If the amount of storage space being used on your old iPhone is greater than the amount of available space in your new iPhone, that would cause the issue you're experiencing.
Review this article for additional information:
Move from your current iPhone, iPad, or iPod touch to a lower capacity device
We hope this helps.
Have a great day!