There is nothing you need to do to get the money. Since you have already filed the 2020 tax return, the IRS should have your most current information.
But, the 3rd stimulus and the additional amount for your child in the third stimulus are going to be sent from the IRS automatically. The IRS is not done yet issuing the checks.
You are right, part of the American Rescue Plan does change 2020. For example, the first $10,200 of unemployment is not taxed, but if you already filed, the IRS will make that correction automatically.
For more information, see: How will the third stimulus impact me?