Well, the biggest problem with meat is bacteria. Bacteria grows in any hospitible climate above 4 degrees c (and will grow below that too tho much more slowly, to a much lower temperature). That's why things spoil more slowly in your fridge
So - if you cannot control the temperature, you must control the environment.
The easiest way to do that is to use an acid such as citric acid or vinegar to make the surface of the meat inhospitable to the little buggers. This is the same concept as 'salting' meat.
In cool weather for a few days (like maybe 15 degrees above) we've had success wiping game meat down with a solution of 20 percent vinigear and water. This has two effects - it makes it difficult for bacteria to grow (which keeps flies off of it , and because vinegar evaporates more easily (which sucks heat out of the meat if you remember your high school physics) it also allows the meat to cool down much faster and thus be more tender.
It is ESPECIALLY important to pay close attention to the neck if you've cut off the head. The blood will turn faster than the meat - then the 'rot' spreads thru the arteries and veins like lightning and even tho the meat itself is good, the animal becomes inedible.
We often mist the game bags with a little vinegar after that to keep it cool and keep the bacteria down.
Another choice which works well is citric acid - available in powder form from most pharmacies.
A strong mixture of this sprayed on the animal will make it VERY difficult for bacteria to form. Unlike the vinegar - it is persistant and works longer as near as i understand.
What you want is a low pH level (acid is low pH - the lower the stronger so to speak). Most bacteria and viruses die in pH levels below 4.6. A lemon would have a pH of about 2.4.
So giving the animal a good coating of citric acid (and you can test with litmus paper the level of acidity) and letting that dry can make the animal hard for bacteria to grow on indeed. Just make sure you get it everywhere
None of that will buy you more time than a refrigerator would - but that should be good for several days in low to moderate temperatures (especially if the animals are hung in the shade and where the breeze is blowing, and if it gets cooler at night so much the better).