I haven't stayed at the Elgin Hampton Inn because I live in Elgin but I did stay at the Hoffman Estates HI to complete the amount of stays that I needed at a certain point last year. I wouldn't recommend it. I had problems to get the continental breakfast, the person at the front desk was rude, they couldn't give me a receipt because they had a power problem that day. As a consequence, the stay was not credited. I was supposed to receive double miles because they had a double miles promotion. I had to call the Hhonors desk to get everything right.
To answer your question, I don't recall a Chicago style pizza place in Elgin close to the Hampton Inn but there's one in the next town, Dundee (The HI is in the border of the 2 towns). Take route 31 N to Main Street and turn to the right. You will see it some blocks to the left on a corner. By the way, this is a very beautiful historical town. There was a Hollywood film shot there last year.
FYI- The Grand Victoria Buffet is the best casino buffet in the state. In case that you need more information, let me know.