I'm not sure how different distro's handle the Docker installation, but here in my office we use RHEL7. When installing Docker, the daemon doesn't start automatically, we have to manually enable and start it. This may explain why you think you need to reboot. It may just be as simple as starting the daemon instead. For me on RHEL7, I just run this:
sudo systemctl start docker
sudo systemctl enable docker
Then I'm set (well there's much more we do, but in this context that's it). But again, I'm not familiar with how other distro's handle it, so some research may be needed for your distro.
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