There is no clear "cut-off date"; and even if you did file your tax return before they make the determination, there is no guarantee that they will have processed it by the time they make the determination.
If they send you a 3rd stimulus check amount based on your 2019 tax return, and you would qualify for a bigger amount based on your 2020 tax return, there is a provision in the law for them to send you an additional check for the difference, after the "additional payment determination date", which is 90 days after the 2020 tax return due date.
See the text of the American Rescue Plan Act, section 9601 which deals with the 3rd stimulus payment. It adds 26 USC 6428B. See subsection (g)(5)(B), for the additional payment:
(B) ADDITIONAL PAYMENT.—
(i) IN GENERAL.—In the case of any individual who files, before the
additional payment determination date, a return of tax for such
individual’s first taxable year beginning in 2020, the Secretary shall
make a payment (in addition to any payment made under paragraph (1))
to such individual equal to the excess (if any) of—
(I) the amount which would be determined under paragraph (1) (after
the application of subparagraph (A)) by applying paragraph (1) as of
the additional payment determination date, over
(II) the amount of any payment made with respect to such individual
under paragraph (1).
(ii) ADDITIONAL PAYMENT DETERMINATION DATE.—The term ‘additional
payment determination date’ means the earlier of—
(I) the date which is 90 days after the 2020 calendar year filing
(II) September 1, 2021.
Paragraph (1) here refers to subsection (g)(1), which provides a stimulus check based on the person's 2019 tax return. Subparagraph (A) refers to subsection (g)(5)(A), which provides for the person's 2020 tax return to be used for those who had filed by the initial payment determination date. So this basically says that, for those who have not filed their 2020 tax return by the initial payment determination date, but did file their 2020 tax return by the additional payment determination date (90 days after due date of 2020 tax returns), they will receive as an additional payment, the difference between what the check would be based on 2020 tax returns, over what they got based on 2019 tax returns.
Of course, if you qualify for even more based on your 2021 tax returns, you can claim a tax credit for the difference on your 2021 tax return (similar to the first two stimulus checks where you can claim a tax credit for the difference on your 2020 tax return).