SSPX:
Our Lady Immaculate
410 Washington Boulevard
Oak Park, IL 60302
Sunday Masses: 7:30 (Low) 10:00 (High)
Priest: Rev. Fr. Charles Ward (I believe) and several assistants
SSPV/CSPV:
Sacred Heart Mission
No address given (call 708-928-1705)
Priest: Various Priests of the Society of St. Pius V & Congregation of St. Pius V
Note: This is a mission that I believe is normally in a hotel or a parishioner's home
Independent:
Saint Hugh of Lincoln
2401 S. 12th Street
Milwaukee, WI 53215
Priest: Rev. Fr. Charles McGuire, Rev. Fr. Stephen McKenna, Rev. Fr. Vili Lehtoranta
Note: While this is in Milwaukee, not Chicago, I listed it since a previous reply did too. This would be a great church to visit even occasionally though, since for a very small church they have a lot of devotions and ceremonies throughout the year including Holy Week ceremonies, outdoor Rosary and Corpus Christi Processions, 40 Hours Devotion, and a special devotion to the Holy Face. They are also literally just down the street from the Basilica of Saint Josaphat, which is worth a visit just to admire its goregous interior because it is, in my opinion, the most beautiful Catholic church in the United States that I have been in (and I have been to around 300 in the US alone!)
Hope this helps.