You do need the iPad to transfer data between your old iPhone and your new iPhone. You just need both iPhones and perhaps a computer depending on the method that you choose to use. See below for more information.
Transfer Data from old iPhone to new-
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
Axel F.