You may not know it, but you have already answered your own question. Most states with elk have their archery and mybe their muzzleloading season during the bugle period. The rifle seasons normally are held after the rut. Now since the time of the rut fluctuates from year to year, occasionally the rifle seasons include the rut. So, if you want to hear them bugle you need to hit the rut which means early (mid-Sep seasons for the most part).

I can't speak for the rest of the people but I prefer to either hunt the muzzleloader season in my native CO if I want a bull and one of the late seasons if I want a cow. Since I'm normally more interested in the meat, the later seasons appeal to me. But then I grew up in elk country so I was always able to hear them bugle even if I didn't have a tag.