The "3rd stimulus" is only a "continuation" in the sense that it was a campaign promise. The 3 stimulus payments were created by three separate laws. A child born in 2021 is only eligible for round 3.
Round 3 stimulus payments are only being made based on 2020 tax return information, there is no way to enroll your child for a stimulus payment. Instead, you can claim a $1400 recovery rebate on your 2021 tax return. See the FAQs on this page. https://www.irs.gov/newsroom/questions-and-answers-about-the-third-economic-impact-payment-topic-b-e...
Your child will also be eligible for the enhanced child tax credit ($3600) and the advance payments. However, the IRS site to enroll your newborn is not ready yet. If you don't get the full amount of the advance payments, the child tax credit will be reconciled on your 2021 tax return. Getting the advance payments doesn't change how much you receive, just when you receive it. This is the site where you would manage your advance CTC payments, although the function to "add a newborn" is not ready yet.
https://www.irs.gov/credits-deductions/advance-child-tax-credit-payments-in-2021
Your child will need a social security number before you can receive the recovery rebate or enroll in advance CTC payments.
