Amending federal after submitting state

I already received my refund for federal but they had rejected my initial state. It turns out I have to pay a dumb tax on a scholarship I got for university. Since my state income tax will increase significantly, can I then go and modify my federal?

It’s kind of circular but this would increase my federal itemized deduction which would then (hopefully) give me a better refund. Is it even possible to receive a second refund?

What sort of documentation would I need for this amend?