Not sure if you’re looking for a faster build in general or if you want a solution other than stopping and recreating the running production container from a newer image.
For the second point, I’ve found that using Amazon Web Service’s ECS platform for deploying Docker apps has been rewarding for our team, albeit a tad expensive. This is because it uses the Blue-Green deployment strategy by default, where you deploy the new Docker container to your production environment and it gracefully updates all containers in your “cluster” (there’s a bit of a learning curve with anything AWS, but in this case that’s just a load-balanced set of instances running your Docker app). Although this is an AWS-specific solution, a Blue-Green deployment method can be implemented on any set of load-balanced VPS’s with the right scripts!
As for the first point, we’re running CircleCI as our CI solution and have found that the build step can be useful in identifying bugs or issues we didn’t see initially. It won’t push the build live if it fails, so it’s an added layer of protection.
Hope this was helpful ¯_(ツ)_/¯