Originally Posted by 747FC

I know OP was curious about First Class mail, but some may not know that UPS and FedEx have their own jet fleet to bring their parcels to the islands:

https://www.ups.com/aircargo/using/services/services/domestic/svc-aircraft.html

https://www.ups.com/aircargo/using/services/services/domestic/svc-aircraft.html

Furthermore, Aloha Air Cargo has a dedicated fleet for interisland cargo transport:

https://www.alohaaircargo.com/about/

And Hawaiian Airlines Cargo ships both interisland and transpacific:

https://www.hawaiianaircargo.com/