The first 3 months of the following year do qualify - however - since the college is keeping it in tune with scholarships, she could end up will all scholarship her senior year and a job to bump up the tax rate if you keep claiming those 3 months ahead. You can claim the tuition and all required expenses paid during a tax year for that year or the following 3 months. If the IRS audits your education expenses, just send them bursar records and your payment records along with any other extras. That department of the IRS functions well.
See page 12, What Expenses Qualify 2021 Publication 970 - Internal Revenue Service to verify your ability to claim all tuition.
Depending on your income, you may be able to qualify for the AOTC. If necessary, you could push some income to your daughter. Depending on her income, it may not be taxable at all. The standard deduction for a single is $12,400 so only income above that is taxable to your daughter. I am also going to recommend you look at another of my answers for help.
AOTC & Lifetime Learning eligibility
$80,000-$90,000
$160,000- $180,000 MFJ
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