The number one thing you can do to keep everything colder in your cooler is to keep keep your ice intact. And the number one way to do that is to build an added layer of insulation.
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Custom cut pieces of foam insulation work well, but an even better choice if you can find it is flexible foam pad (like the backing on mouse pad) that can lay across the top. A custom fit allows you to lift the lid, and then only the end of the pad to get what you need – thereby not exposing the entire cooler to warm air.
And the more insulation, the better, just make sure it’s as snug a fit as you can get.
If you go for two layers, cut the inner layer in half and glue it to the single top layer. This will do best to build a “hinged” lid to keep the warm air out, and the coldness in.