You are not going to need the old backup to restore your old iPhone, This is because the Quick Start method does not wipe the data from your old iPhone. The data on the old iPhone remains as it was.
Transfer Data from old iPhone to new- There are two different methods.
Do you have a backup created either on iCloud or on your computer?
Method 1 - Backup your old iPhone on your computer like this ➔ How to back up your iPhone, iPad, and iPod touch - Apple Support and then Restore your new iPhone from that backup like this ➔ Restore your iPhone, iPad, or iPod touch from a backup - Apple Support
Method 2 - The Quick Start method and here is how to do that ➔ Use Quick Start to transfer data from your previous iOS device to your new iPhone, iPad, or iPod touch - Apple Support
Note that when you transfer data using Method 2 then you do not lose any data on your old iPhone. The old iPhone retains the data that it already has and just transfers/copies the data to the new iPhone.
Axel F.