Nolo.com has the requirements for collecting unemployment. Some excerpts:
In Illinois, as in most states, the base period is the earliest four of the five complete calendar quarters before you filed your claim for benefits. For example, if you filed your claim in October of 2012, the base period would be from June 1, 2011 through May 31, 2012.
So if you are filing a claim today (July 22nd, 2019), the base period would probably be between April 1st, 2018 and March 31st, 2019 (if I got the calendar quarters right). That's the period when your income would be established. If all your income was from April 1st, 2019 and later, you might not be able to collect any benefits.
You must be out of work through no fault of your own to qualify for unemployment compensation. If you are laid off, lose your job in a reduction-in-force (RIF), or get downsized for economic reasons, you will meet this requirement.
So you may have to lay yourself off formally in some way. But you seem to meet this qualification.
To maintain your eligibility for unemployment, you must be able to work, available to accept a job, and looking for work. (See Collecting Unemployment: Are You Able, Available, and Actively Seeking Work? for more information on these requirements generally.) If you are offered a suitable position, you must accept it.
You must register for work with the Illinois Employment Service, either online or in person. You must keep a record of your job search efforts, including the dates and places you apply for work. When you apply for benefits, you will receive a form on which to record this information, which you may have to provide to the agency.
To be clear, you have to look for another job, not at your S-corp. If offered such a job, you have to take it. You can't simply look for work as a contractor. In fact, looking for that work won't count at all towards the requirement that you look for work. It would be in addition to your mandated search.
All that said, if you find another job (not the S-corp) and then find work for the S-corp, you can quit the job and go back to the S-corp. The point here is that you can't keep getting unemployment once you are offered a job.
Unemployment costs may be higher
Another warning. According to the Illinois Department of Employment Security:
For experience-rated employers (those with three or more years of experience), the contribution rate is based on a ratio, called the benefit ratio, which is determined in such a way that the greater the unemployment caused by the employer, the higher the rate.
So collecting unemployment may cause you to pay a higher tax/fee in the future. As I understand it, this would be for all employees, not just yourself.