Spot and stalk would be the longest odds, less chance than any other method I can think of. Yet I have a friend who hunts cougars that way on Vancouver island where there is probably the heaviest population of cougars on the planet (and they are still scarce). He knows certain ledge areas where cougars often lay, picking up some sun, and close to lots of deer. He's seen several that way but the only one he killed was not spot and stalk. He saw it on a ledge at 60 feet when he was hunting deer.
A knowledgeable woodsman told me to watch for cougars sunning themselves in the tops of bull pines on cold mornings. I have looked quite a bit and I think have seen one, while hunting deer in the mountains west of Wenatchee, WA. With bincoulars I spotted a tawny patch apparently lying on a big limb near the top of a huge old Ponderosa pine on the canyon side opposite me, half to 3/4 of a mile away. The sun was rising over my left shoulder, lighting up the far side. Whatever it was never moved as I glassed for deer, moved to another spot, etc. and checked on it once in awhile. It somehow looked alive, with that sense hunters get when light is on fur rather than wood or stone, but I decided it was inanimate. Then suddenly it was gone.
If a man had unlimited time and interest, and was fit, it would be a fun quest to get a cougar that way. You'd learn a ton about all kinds of game with that much time afield looking.