Perfect Cougar Pelt
The cougar spawns in multiple places around the map in Red Dead Redemption 2, so the best way to find one before it finds you is to listen for it's calls and stop moving until you figure out where the kitty is hiding.
One tactic for finding a cougar is to use the buddy system. Since cougars are expert mousers that can settle in and wait for as long as needed, it can be very difficult to see them even with Deadeye, and their trail fades quickly when they go to ground. So, using the buddy system, you can let other travelers precede you into the cougar's territory and watch for it to go after them, then kill it while it is either returning to its resting spot, or while it is feasting on the poor person you used as bait.
There are also events in a few places around the map where a cougar has treed a fellow, and you can get some honor points for shooting the cougar before the man slips out of the tree and gets eaten.
Always aim for the head of the cougar, so you kill it in a single shot and do not damage the pelt.
Roanoke Valley Cougars
One of the best places to hunt for cougars is in the Roanoke Valley region, since there are several spawn locations around this area.
Traveling north on the western side of Roanoke Valley you can sometimes find a cougar up a shallow ravine where the road heads back down to a bend in the river. You will want to be careful approaching the area and watch out for the cougar to be behind one of the trees looking toward the north-east.