How to automatically pull and deploy updated Docker images

We want our test and production stacks to be automatically updated every time something new is pushed to the test or release branch. CI builds the docker image on successful test runs, then stores it in our private registry. But how do you automatically pull and deploy those updated images? I looked into the Watchtower project, which is interesting. You add Watchtower to the stack, and it will diligently check for new versions of the images used by the containers in the stack, pulling, building and deploying as needed while the stack is up and running. »