If you paid into MO, that does not go to IL. So if your tax liability for IL is greater than your liability to MO, then you would still owe taxes to IL.
In general, your home state will tax all in come from all sources. What you will do is you will file a nonresident return for the states that you do not reside in, but you visit for work and earn money in. When you file, you will fille out your nonresident states first. Then you will fill out your resident state, claiming a credit for taxes paid to the nonresident states on your home state return. You may or may not still end up owing money to your resident state depending on whether or not their tax rate is higher or lower than your nonresident state. If the non resident state had a higher tax rate than your resident state, your credit will be limited to the amount of tax you would have paid to your resident state. They will not give you a refund of the taxes you paid to the nonresident state.
Multiple States
File Non Resident State Return
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