There isn't really anything to "recoup" just yet as there is no stimulus just yet.
You should amend your return, as it is the right and correct thing to do and you may receive some additional refund for 2020. Depending on other items on your return you could also potentially have losses you could carry to 2021 which might help you then.
However, if the current bill is assigned into law this week, as some expect it may be, your amended return will not likely be processed by the time that they start issuing payments and you would be left to claim it in 2021 based on your 2020 income.
One reason for this is that the amendment form isn't ready yet in TurboTax, and won't be for at least another 10 days. I expect it will get pushed back further given that the law as currently proposed has retroactive tax changes that software companies like TurboTax will need to put into their programs so people can file their first returns before they focus on amendments.
Also, amendments, even when e-filed take up to 16 weeks for the IRS to process. (Currently, mailed in returns are taking much, much longer than this - they haven't even finished 2019 yet!) So while you could amend before TurboTax is ready, it would slow things down considerably.
