I was using a very simple deploy script that just ran the aws command line tool, and then made a simple curl request to invalidate the cache in CloudFlare. I've seen Netlify mentioned a lot, and I was about to say that I don't really need it...
But I just signed up, and I was immediately convinced to switch over. Their sign up flow is amazing. It really is like a Heroku for static content.
The main thing is that I don't have to run my deploy script anymore, and I just `git push`. I could set up the same flow with GitLab CI, but this is just a much nicer service, and it's free. And they do cache invalidation automatically. I don't know how their CDN compares to CloudFlare or CloudFront, but I'm sure it's good enough.
P.S. Another thing to mention is that I'm actually generating a static site from my Rails app. So far I really like using Rails helper methods and libraries to generate my static landing page, such as the recaptcha and stripe gems. I'm using "stripe-rails" to define all my plans with a DSL in config/stripe/plans.rb, and then that automatically generates my pricing page HTML. Then I just run `wget --recursive` to download the root page and all of the links.
I have my root domain record pointing to a CDN where I host the static files, and then the sign up or sign in links point to the Rails site running on a subdomain, e.g. app.domain.com. I really love this setup, because no matter how much traffic your site gets, the landing page will never go down.