The existing functionality is quite limited in the support of delivery services around the world.
I propose the idea of integrating https://www.17track.net/en/api to support 2500+ carriers and 190+ airline cargo across the world. A user might search with "Tracking Number GLS", "GLS tracking number" or "tracking number". This provides the user a convenient way of tracking their upcoming parcels.
Otherwise you may integrate more delivery services across the world. It would be great to learn more about which ones are currently supported.
Delivery services I couldn't find with your system:
Do note that some users don't want the number to be shared with the service. That would require a special user setting in place.
Some APIs require the user to send their postal code and city, perhaps Kagi can find a solution for that too.