Note: I would read all the way through as there are two viable methods.
For this achievement I loaded up the Northwest Region, Paddlers Bog area, open hunt. (Also a great place to grind out some moose kills to unlock the moose hunt.)
Equip the 12x scope on the .338 Mag with the Polymer Tip (ammo you have to purchase), Scent Mask, Moose Call, and 10x Binoculars (or greater if you have them.)
Start the game and use your moose call when facing N/NE to check for a moose. If there is one then you are in good shape. You may need to use your moose call to lure it over the ledge a bit. Then mark it Default (A). Once you have just start heading in the opposite direction until you can spot the moose from 250 yard out. I would spot it out to 255 just to be safe. Don't worry about spooking the moose that are south of you as they should not effect the moose you're hunting. Make sure you hit your vital of choice.
Hopefully a male moose (can be any big game, but have never seen any other big game types at this spawn, also not sure if it can be a female or not), will spawn North/Northeast of your position.
You can try the same trick the moose that spawns to the South, it may be easier. I was able to line up a shot for over 250 yards, but I still had to call the Moose a little bit. I would probably try this one as this one first as this Moose always seems to spawn.