I live in Illinois and have recently remarried. Can my husband's ex-wife go after my income to increase her child support payment?

Usually the answer to that is no. In California the answer is no- almost never (there is an exception to the general rule in our Family Code but its application has never been discussed in any published case opinion and family law experts report to have never heard of a case where the exception has been applied by any trial court in the state).

I'm not an expert, but where I live the answer is no.

Here, child support is based solely on the income of the biological parents, not the combined household income. It's a scale that is pre-determined based on the amount each biological parent earns. They do not adjust the payment should either party remarry. The only time the support payment is adjusted is if one of the biological parents loses their job, changes jobs, gets a second job, etc. If the custodial parent wasn't working when the support order was put into effect, the non-custodial parent may petition the court for a modification in support if the custodial parent suddenly gets a job, or if they were working and they get a better job with a substantial increase in income.

So either party can petition the court for a modification and the court will determine the amount based on both of their incomes, without including the income of any new spouses.