I’m the opposite of the above opinions. I live in the middle of several of the most desired goat units and watch/hunt them each year as friends/Family draw tags. Yes goats get truck shy after getting shot at by every road hunter since opening day. Yet the universal go time for elk is mid October for most the state. I have yet to have seen a antelope hunter during the last couple weeks of the season which is my favorite time to chase goats. Additionally if your unit has private ground, most ranches will have deer/elk hunters on them which again pushes goats back to public.
Another reason I prefer the last two weeks is they are typically herded up in large pre winter groups so spotting them from a distance is a breeze. It also helps in determining trophy size when your looking at 10-30 bucks in a group (I hate judging a solo buck).
I typically tag out elk during archery so I have friends come mid to end of Oct and we always have a no pressure public land hunt with a crazy amount of bucks. Winter weather is a plus in my book as again, it just herds them up more. Just remember this is antelope, you can kill one any time of the year. I’ve never once heard of someone drawing a goat tag in WY and not filling it if they actually got their butt out of the truck and walked. It’s hunting, not a McDonalds drive through.