First off...

I shoot most of my deer on evening hunts, so by the time I get the deer dressed out and checked in it's usually late.
The deer sits inside my covered pick-up truck bed until the next day/night when I bring it to the processor.

All I do, is buy 7 bags of ice....
Three for the chest cavity, one under each shoulder/hind quarter, and 0ne on top of each shoulder/hind quarter.
The ice packs will basically do the same, but more may be needed, due to their smaller size...