Run:

$ docker run --detach --net=host --cap-add=NET_ADMIN --cap-add=NET_RAW --volume /var/run/docker.sock:/var/run/docker.sock quay.io/dreae/docker-external-ip

After that, if any other Docker container has an environment variable SERVER_IP set, with an IP address to use for containers external IP, iptables will be configured to route container's traffic from that external IP.

A chain named EXTERNAL_IP is created in the nat table into which all the rules are added. And one more empty chain is created after this one for any additional custom rules you might want to add, named AFTER_EXTERNAL_IP.