There’s no GitHub way (small lie, see below), but there’s also nothing to fear.
By definition, your fork of a fork is a fork of the original.
When you open a pull request, you get the option to choose both the origin and the destination for your pull request. The choices available there obviously depend on the fork graph, but as long as there is a path in the graph between the 2 repositories, you should be safe.
Also, since pull requests live on the website side, you don’t even need to add a remote as long as you don’t want to use it from git.
Now of course, you might want to reconsider your place in that graph, and make yourself a direct child of the real upstream, but that’s mostly unrelated.
As said earlier there is actually a twisted way to have multiple forks, which is to create organizations and fork in them. That way you can “own” multiple repositories in the same graph. But there’s really no need to go there.