The way you’ve stated, docker tag ...; docker push ... is the best way to add a tag to an image and share it.
In the specific example you’ve given, both tags were in the same repo (myregistry.com/myimage). In this case you can just docker push myregistry.com/myimage and by default the docker daemon will push all the tags for the repo at the same time, saving the iteration over layers for shared layers.
You can use the same process (docker tag ...; docker push ...) to tag images between repositories as well, e.g.
docker tag myregistry.com/myimage:v1.2.3 otherregistry.com/theirimage:v2
docker push otherregistry.com/theirimage