Ideally, you are scouting all year with focused scouting happening at strategic times.
Those times are something like:
1) 10-days before season opener so you are DIALED about where elk are spending their time/rotating. Do a little listening at night for early bugling. Look for rubs. Be ready for the opening day assault.
2) During the season, always be scouting. Rubs that weren't there yesterday. Bugles at night as you get off the mountain. Fresh trails and scat -- if it's fresh, not even the flies will have found it so it will be smooth on top and not have a bunch of little holes where the bugs ate. A lot of guys will say, "Hey look! Fresh sign!" And that scat has been there a week. Consider the flies.
3) Immediately after rifle season ends (or whatever the last legal bull season is in your area). This will give you an idea of what bulls carried over. Ideally, you have trail cams out and don't even have to scout. Bulls are tired and laying low so "scouting" is mostly cams and perhaps jumping bulls if you are quail or grouse hunting for example.
4) Spring calving season. Why? Because knowing when a calf drops can tell you when its mom was bred. Seeing lots of newborn calves can tell you when lots of cows were being bred -- i.e., peak rut. So be in the woods around June and watch for those calves. An elk's gestation period is 240-262 days. Let's say you see lots of calves on June 1. 262 days before then was September 12. Amazing. Watching this trend over time in your area can give you a pulse on when peak rut occurs, and also important, if any pre- or post-rut mating is happening.
I am writing as if you live within a reasonable distance to your hunting grounds. I do, and we soak cameras year round with focused soaks starting in mid-June on the Oregon coast.
If I had to travel somewhere to elk hunt, I would arrive at least 3 days before the opening to scout hard for that time period and be ready to kill on opening day.