I've been going to unit 62 periodically since the mid 80s. Every time I've went except one, someone in our party has killed elk. Usually we are close to 50% successful for the party, and you can hunt mule deer and bear. I like to go later in the season because there are less people and the elk are more active. Also it can be downright hot in the early season - I don't like to hunt when temps are in the 80s. Find a ridge that goes out from the divide road that easy hiking is blocked by a private ranch. Go down in the canyon to go around the private property, come back up on top and you will likely get into undisturbed elk. Probably have to hike 3-4 miles with a lot of elevation change to be a mile in. You will be more likely to see cowboys on horses and cows than other hunters or ATVs once you get back in. It will be tough getting the animal out though so be prepared to bone the animal out (learn the gutless method) and for some difficult packing if you don't find someone with a packhorse.
I also want to add that the private land is probably leased to hunters or outfitters and they do not care much for trespassers so get an up to date map of federal lands and know where you are. On the other hand the cowboys you meet are generally really friendly and happy to help you out, especially if you have a beer or two to offer up.