You can grant Cloudflare access for your hosted domain to more than one email address. Each address is scoped to your own domain.
How the invitation works. The invitation arrives by email and expires after a few days. For that reason, support sends it only once the target mailbox is actively receiving mail.
If your new mailbox is not ready yet. If you need access before then, let support know any address you can read right now, and the invitation is sent there instead.