Typically you can check into your hotel ahead of the posted check-in time, but there's no guarantees your room would be ready before that. But it would at least allow you to still use the new hotel's amenities.
The benefit of taking an early morning flight is that you get a full day in once you arrive, and if there are any delays/cancellations, there are usually several more flights going to the same place throughout the day. The con is having to get up early, but you likely wouldn't have to contend with much traffic.
The benefit of taking a late night flight is that you get the full day wherever you're leaving, although you'd still have to check out of your room by whatever the check out time is. You'd be inclined to hit traffic going to the airport so would lose a little time having to leave earlier, and you run a greater risk of your flight having issues if something happened earlier in the day and there was a downstream impact. Plus, fewer flights to move you to later in the evening in those cases.
We left the Kaanapali region of Maui, about a 45 minute drive to the airport around 10:30am for a 1pm flight (pre-Covid). Leaving at 10:30 gave us just enough time to pack up and grab breakfast, but not much else. By the time we got to Kauai, it was about 2:30 and then we had to get luggage, pick up the car, drive to the resort, check into the hotel, unpack... it was probably around 5pm by the time we got settled. We had a nice dinner, but again, no time for anything else.