My trick for cooling down a large pot of chili, gumbo, pasta sauce, etc. is to set the pot into a clean sink (drain closed, of course) and then fill the remaining space in the sink with ice cubes and cold tap water. This will generally take the heat of the pot and the ingredients down rapidly, though it might take two rounds of ice and a little time. I usually then transfer the ingredients of the pot to smaller containers if I can't fit the pot into the refrigerator. I've yet to have this technique cause any detectable spoilage of other foods in the fridge, big pot or not.
You could use the same trick by putting, say, a leftover piece of hot roast into a Ziploc bag, removing as much air as possible, and floating the bag in ice/water in the clean sink or a bowl. Thence to the fridge, even if the food is still slightly warm.