I just did this job search last year.
The hospitals on the East Side that hire new nurses are primarily Gateway Regional Medical Center and Alton Memorial.
Alton Memorial is a BJC facility. They're decent enough to work for.
Gateway is owned by Community Health Systems. I left after four months. I wouldn't recommend it as a first experience. Other people in my nursing class started there and left just as quickly as they could.
St. Elizabeth's in Belleville has a nursing residency, but it's very tough to get. Totally worth it to apply there.
Belleville Memorial and Anderson (in Maryville) don't hire new nurses. Kenneth Hall in East St. Louis also doesn't seem to hire new nurses.
I haven't checked out Touchette Hospital in Centreville, and would never apply there.
I don't much about St. Anthony's/St. Clare's in Alton, except they don't have many positions listed on their website.
If you want to go further out, Washington County Hospital in Nashville, Illinois will hire new nurses and Greenville Regional Hospital in Greenville will also hire new nurses.