The problem is that your company has already sent your withholdings to the "wrong" state. The only way to get such withholdings back is by filing a part-year return with the previous state (which you have to do anyway), showing that you don't owe as much tax as was sent to them.
You will have to pay the extra tax to your new state, and hope that there is not too much time before the previous state sends you a refund.
There is no automatic way to do this.
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