Some outside scholarship organizations will have explicit rules that prevent any overage beyond the estimated cost of attendance.
Many colleges will also have rules prevent the same overage.
Your best hope is that the outside scholarship organizations write the check directly to your daughter and don't include the school's name on it. If the school's name is on it, my guess is that it will not result in additional money for your daughter but will help the school that was generous with your daughter.
You could check the Financial Aid office website for both schools to see if they spell out their specific policy. They should do so.
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