gistfile1.sh

# go to your repository
cd my_project

# check your existing remote
git remote -v
# origin git@mydomain.tld:my_project.git (fetch)
# origin git@mydomain.tld:my_project.git (push)

# Add a new remote, a github.com private repository for example
# the --mirror flag is what's different from a simple new remote
git remote add github --mirror git@github.com:Openhood/my_project.git

# Check the new remote
git remote -v
# github git@github.com:Openhood/my_project.git (fetch)
# github git@github.com:Openhood/my_project.git (push)
# origin git@mydomain.tld:my_project.git (fetch)
# origin git@mydomain.tld:my_project.git (push)

# To discover the difference check you .git/config
# the new remote has the config mirror = true
cat .git/config
# ... file start skipped ...
# [remote "github"]
#   url = git@github.com:Openhood/my_project.git
#   fetch = +refs/*:refs/*
#   mirror = true

# Now all you have to do to mirror your entire repository is
git push github