How to determine if meat needs to be aged?

This is not a full answer, and different than aging, but must all meat needs to either be cooked immediately after slaughter or as least be allowed to go through the rigor mortis cycle. Larger animals may take longer than smaller, but it still occurs. Many home raised chickens for instance are very disappointing to the grower not because their is anything wrong with the bird or the way it was raised, but because this step is missed. Immediately after death, normally muscles go to a temporary flaccidity, but this quickly is lost with muscles becoming rigid and causing joints and connective tissue to stiffen. This is true regardless of if the animal is immediately parted or not, in fact parting while in the rigor cycle can cause muscles to contract in ways that will cause it to stay tough even when the cycle is allowed to complete if you cut across muscle.

With a chicken, an item I have more experience with, this process if relatively quick. If you cook the bird before it cools, you are usually fine, but once it stiffens it takes maybe 24-36 hours to complete the cycle. Cooling the bird aids in two ways, it helps get into the cycle faster, and prevents spoilage during the period. Larger muscle mass will take longer to complete the reactions to complete the process. In a human sized animal, I believe the peak of rigor is about 12 hours after death without cooling to speed it to that point, and the relaxation is completed in the 2-3 day range later, or roughly 3 days to complete the process, but temperature will make this vary. A bit of a guess, the I would assume that something like a cow would be up in the 5 day range. My understanding is that in beef electrical current is often applied to the muscles to cause them to contract and relax and speed the process, but I have not personally seen it done.

I say this may not be the answer you are looking for as I would not call this aging. Though there are chemical and physiological reactions going on, this is different than aging meat such as dry aged steaks. That is a process in which the meat has already gone through that rigor cycle and then is allowed to age, often removing some of the water and partially drying the meat to produce a more intense flavor, but I think from your question you are keying more on the rigor cycle.

Beyond letting rigor pass, it depends on the age of the animal.

Keep it whole if you can; less access for bacteria. If you must you can cut it up and age it in the fridge, but that's a hot weather solution. If you can keep the carcass below 40F another way, do. The usual is to hang it in a garage and put a bag of ice in the chest cavity, and maybe wrap it with an old sleeping bag if it needs a bit of insulation to stay cool.