Since you tagged this as cloud, this is my best suggestion:
At this point that one user you created and left out of that Group should not see any Projects.
End result: users in general-access will see all Projects (also the ones created after this change because you edited the default Scheme), the one User you left out of that Group will just see the Project (or Projects) that you have applied the new Scheme to.
Caveat: every new User you add to your instance that shouldn't be restricted to that one Project will need to be manually added to Group general-access.
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