A lot of good conversation about elk and guns ... here's my twist: if you are going to use the .243 like an arrow ... well-placed shot behind the shoulder, catching both lungs and maybe heart, and then out the other side, plus deliver some shock on the way thru ... the 243 would be like a 30-06 125 gr, 150 gr, and so on and so on. After the appropriate amount of time - probably not long - you follow your animal into the thicket to where he or she has died, and there you have it.
But if you have to `put the animal down' - e.g., deliver as much energy (e.g. destruction) as possible into as much muscle and bone as possible, at the probability of losing meat, but the better likelihood of getting (or not losing) the thing - then the 243 is going to be much different than the bigger stuff.
Part of hunting is learned beforehand ... the other in doing.