No, GitHub will only let you fork a single time. Once you have created a fork, the button to fork a repository will only link to your personal fork.
Or if you are part of organizations, you will get a target selector, prompting you for whom you want to create the fork; but for every account that already has a fork, you will also just get a link to that fork.
Of course, nothing prevents you from pushing a copy to a different/separate remote, but that will not maintain any relationship between the original repository and your “fork”. GitHub’s fork relationship only exists from the “Fork” button and can only ever exist for a single fork per account.