Do you really mean boot faster, that is, start serviing pages sooner after you restart the container? No. It just takes a while to get rails and all of the stuff up and running.
If you mean the downtime associated with rebuilding a container, the the two-container solution is what you’re looking for.
And if that’s not fast enough and you want zero downtime, then you can do stuff with a load balancer and spin up the new container while the old one continues to serve pages and then shuts down the old one after the new one is ready.