What is the right way to add data to an existing named volume in Docker?

You don't need to start some container to add data to already existing named volume, just create a container and copy data there:

docker container create --name temp -v my-jenkins-volume:/data busybox
docker cp . temp:/data
docker rm temp

You can reduce the accepted answer to one line using, e.g.

docker run --rm -v `pwd`:/src -v my-jenkins-volume:/data busybox cp -r /src /data

You can certainly copy data directly into /var/lib/docker/volumes/my-jenkins-volume/_data, but by doing this you are:

• Relying on physical access to the docker host. This technique won't work if you're interacting with a remote docker api.

• Relying on a particular aspect of the volume implementation would could change in the future, breaking any processes you have that rely on it.

I think you are better off relying on things you can accomplish using the docker api, via the command line client. The easiest solution is probably just to use a helper container, something like:

docker run -v my-jenkins-volume:/data --name helper busybox true
docker cp . helper:/data
docker rm helper