Hi FCC_Riemslag,
We understand when you are wondering when you should transfer your SIM card from your iPhone 6 to your iPhone 13 mini. If the SIM card is compatible, you'd want to transfer it before your new iPhone tries to activate your cellular services. Your iPhone won't be able to activate to a carrier without a compatible SIM card.
If you'd like, you can read more here: Set up your iPhone, iPad, or iPod touch - Apple Support Specifically step number 3 as seen below:
You need to connect to a Wi-Fi network, a cellular network, or iTunes to activate and continue setting up your device.
Tap the Wi-Fi network that you want to use or select a different option. If you're setting up an iPhone or iPad (Wi-Fi + Cellular), you might need to insert your iPhone SIM card or iPad SIM card first.
Get help if you can't connect to Wi-Fi or if you can't activate your iPhone."
Take care.