Glad you got around to mentioning the altitude.
Here's the deal about meat spoiling, Water molecules are the most dense at 37.5 degrees F. You set your refrigerator to cool things at about 38 degrees, because the dial is numbered in 2 digit increments. This is the best temperature to keep meat from spoiling, as bacteria are the least active at that temperature. Below 37.5 degrees, the bacterial activity increases as the water molecule also increase in size. Above that temperature bacteria become more active. The optimum temperature for rapid bacterial growth of the kind of bacteria that spoils meat is between 50 and 85 degrees F. Above that and the meat begins to cook and kill off bacteria. Below that the bacteria cannot be very active-but, they will still spoil the meat if given enough time.
NOW, altitude plays a big role in what you can do and what you can get away with. The higher the altitude, the thinner, and DRIER the air. Bacteria, like all living organisms, generally need heat, food and water to survive and reproduce. Its the process of reproducing that leaves toxins behind in the meat that "spoils " the taste. When you get above 6,000 feet, you are above the most humid zones, and as you go above that elevation, the air becomes thinner, dryer, and Colder longer. Bacteria die over night when the temperatures drop below freezing, or at least into the low 30s. They cannot multiply and reproduce at high altitude because they don't have enough warmth, long enough, to get the job done. They will multiply and spoil meat if you get that rare day when its in the high 60s, and 70, or hotter, however.
You can get away without ice at high altitude if you:
Keep the meat in Shade- where its cooler, but the cool, dry air still circulates.
Keep the meat dry, and clean. Dirt is the vehicle that brings soil born bacteria to the meat. Keep the debris, and dust off the meat. Cover it with something. Many hunters still leave the hides on. I don't recommend it, because you can get spoilage under the hide. Its a terrific insulator, and it holds heat in long after the animal is killed and even field dressed. Use rags, towels, paper toweling, whatever to remove the blood and any other fluids on and on the meat.
I recommend using Cheesecloth to wrap the meat, after removing the hide. Cheese cloth is cheap, available in the automotive section of the big discount stores for waxing and polishing car finished, at a price that is a third of what the same stores charge over in the " linens and fabrics" department. Check it out. The Cheesecloth I have bought is made in a cylinder, like a large ladies stocking. Only these can be expanded wide enough to cover the entire chest and rib section of a deer or Elk carcass. The Cheesecloth will keep the meat both clean and dry.
Keeping it cool depends on where you are and what you have at hand. The idea of coolers, with containers of ice in them is great. You can usually get permission from Motel operators to put your Ice containers in their ice maker chest over night, so they are freshly frozen the next day, when you are driving a couple of days to your hunting area. I buy Bags of Ice at every tavern, Gas station, truck stop, Stop and Rob, etc. that I see. Break up some of it and put it in zip lock bags, and put that inside the carcass, and in garbage bags ( ice chests if you have them) with the meat, to keep the meat cold, but not frozen.
Those large plastic, multiple ply garbage bags make a great substitute for ice chests when you don't have them. Put more bags of ice over and on the sides of a bag containing the carcass, and then cover it with a wool blanket, or sleeping bag, to keep the cold down against the meat in that bag. The bags will keep the meat cool, protect if from road dust, and odors, like diesel fumes, and from the heat generated by your vehicles engine.
If you have to travel more than a day to get home, simply refresh your ice with more ice every day. I check the status of my ice and meat every twelve hours, and buy more ice if needed.
A lot of people decide not to transport bone, gristle, fat, and sinew they can't eat, and can't use. So, they bone out the meat in their hunting camps, and simply bag up the muscles that are saved. It certainly cuts down the amount of an animal you are hauling home, and makes it quite possible to fit most if not all of it in the available ice chests for transporting home. I fold up the hide of my deer, and bag it, then put a bag of ice inside the plastic bag, to keep the hide from spoiling, and the gnats, and bacteria at bay. If you are not going to use the skull or head, dispose of them away from your camp. Coyote, ravens, eagles, and all kinds of lower mammals, and bugs, will quickly consume the scraps, and mice will eat the bones for the calcium. Any scraps or meat that can't fit in the coolers can be kept in plastic bags, with ice bags inside, until you can stop and buy another cooler(s) to put the extra meat in.
Experienced hunters haunt garage sales, and estate auctions to buy large ice chests- the metal sided Coleman coolers last forever, it seems-- so they have enough to transport a lot of meat. And everyone has sharp knives, hones, and for some, even a saw to help in the butchering.
I don't personally use a saw, because I don't want saw chips of bone being cut and the driven into the muscles surrounding the bones. Its another way to introduce bacteria where it doesn't belong, and it will spoil the meat. I use the scraps from kills to make sausage. The recipe is on this forum down below on the index page.
There are places to hunt in Colorado where its NOT cool enough, and its not HIGH enough to have cool, dry, air long enough each day to prevent spoilage.
I am not in favor of the " Casing " mentioned, that some like to put on their meat. Its hard to cook, and harder to eat. I don't want to waste that much meat, thank you. In warm areas, locate a source for Ice cubes, in bags, and plan on trips when needed to get more ice.
I know one party who hunted during some fairly warm weather out there, but they found an overhang where a snowdrift still existed in August, protected from the heat of the sun. They put their Elk and Mule deer meat in the snow bank during the day, to keep if from spoiling. If you take a modern camper out there, these often have refrigeration units in them that can make a small quantity of ice each day. Talk to the dealer about improving the ice making capacity, with any add-on devices. Its not the first time the industry has faced this issue. They have units that are installed on refrigeration semis all the time. You won't need anything that large, but something smaller to make ice, or allow you to keep your coolers and ice containers frozen.