I can't answer your question specifically as it applies to a Hunter but I suspect the ice box outside insulation may be the same as on a Catalina 30. My ice box was great at melting ice and I gave up on it and used it for storage BUT one day I was poking around near the ice box and was amazed at the thickness of the insulation around 3 side and the bottom. The problem was the 4th side next to the hull, which had zero insulation, and also the lid which had zero insulation. I purchased a small sheet of insulated board and placed it inside the ice box on the 4th side and on the lid.
That simple and inexpensive mod improved the the efficiency of the ice box by 100%. Additionally, wrapping the ice in brown paper super market bags helped also. So, to answer your question, I think it's an easy, inexpensive and quick mod and worth the small effort. Eventually, I installed a small fridge and the ice box was converted back to storage.