I guess a lot depends upon the area you are hunting but where I hunted in the Bighorns the bulls don't hole up in deep dark areas compared to possibly other regions in Wyo where there is deep, dark timber. The shiras I hunted in Wyo were on the run searching and smelling cows the day after they shed their velvet! I watched and called in a gob of bulls early. They would search for a cow.....go up and smell them....and head on to another cow. Once cows go into estrous the same bull will likely stay with the cow.
When bulls are on the move searching for cows they are more than willing to come into calls. I do more cow calling and whacking branches on trees than bull grunting. Bulls can hear cow calls and branch snapping a lot further than bull grunts. Sometimes it may take a while for them to travel because they can hear these from long distance. If you think about it, a bull's antlers are almost like a mega-phone bringing sound to their ears...and they have excellent hearing!
With that said, a lot depends on the area you are hunting, weather, etc. Moose don't like hot weather so that may slow down the rut and their activity. Cooler weather may prompt them to start searching for cows! My preference is to hunt them early when they are mobile vs trying to find them when they are less mobile staying with an estrous cow. Obviously once a cow is bred the bull will be looking for another cow.
I certainly wish my season would have started earlier because I saw bulls all summer long in the same areas. Once they shed their velvet they totally moved miles from where they hung out all summer. I knew this was going to happen and luckily I knew approximately which directions they moved. It was super important to take advantage when you saw them after they shed velvet because they could be 5 to 10 miles away the next day.