Get the animal quartered, boned out and hung as soon as possible. Then get it where it can be preserved. If you have to worry about the temperature you are probably far away from you truck with the ice in coolers. If hunting national forest during bow season it is cool enough to keep hung meat overnight. The biggest thing is to get the bones out of the meat. If you don't have horses you want them out anyway as they are heavy to carry.
You don't need to gut an elk before quartering it. After you get the quarters and back straps just don't forget to go in and get the tenderloins. You can get very bloody gutting an elk and if you are only packing the meat out it is unnecessary to do. The more blood you get on you the more flies get on you. If I could winch the elk into a pickup I would gut one again, but otherwise no.
Also, get good reusable fabric game bags to protect the meat from the flies which will be there if it is over 55 degrees.