Hawaii is a domestic destination. Your original question makes no sense unless you have separate tickets, the first a Basic Economy and the second regular. If it’s on (non-BE) ticket, you have carry on privileges on both segments. If you have a single BE ticket for both flights, you have no carry on privileges beyond small personal item. Maybe you think Hawai’i is an international destination and you’re confused?

Edit: Now I see you confused yourself and then followed up by confusing us

You get normal carry on privileges with an award ticket on both flights. And Hawai’i is still part of the United States and, as such, is still a domestic destination.