Well the obvious answer is during the Rut. In WY I find the best dates to usually be between Sept. 9th to say Sept. 16 these are usually the magic dates for me, however they are certainly subject to change, ie...weather, moon phase, temps, etc.
The next answer has to be time of day. Make sure you are where you want to be hunting well before light and like wise stay until complete darkness, get a good head lamp. This last few minutes of light has produced some of my best success! Like any big game animals this is when the elk will be most active. Stay on the mountain through out the day and sit water or wallows. More time afield eventually leads to more oppertunity.
Good Luck and Good Hunting