I got one of these evaporative dog pads and wrapped it around the outside of the cooler, and would periodically spray it down throughout the day:
You can get blankets, towels, etc all made from the same material, but the dog pad was the right size for my cooler and one of the cheapest options.
Of course, the most important thing is to minimize how often you open the cooler. I pre-bag my beers so each day I move a bag of beers from my main cooler to my day cooler. That lets me pop it open in the morning, grab a bag, and close it right back up really quickly.
I froze half the beers in each bag pre-trip, and those were the ones I drank later in the day. Early in the trip that ended up being too many frozen beers (they were still frozen when I wanted them), and later in the trip there were not enough frozen beers (didn't stay cool all day). Next time I'll try freezing less beers in the early bags and more in the later bags.