I would consider extending the season into spring rather than fall. Have some sort of mini-greenhouse around your fig in early spring to get them to break dormancy earlier. Just have to be careful that you don't have any hard freezes that can damage new growth. Maybe you can set up the greenhouse in early-mid April, you may still get some light frosts but with heat sinks (ex brick wall, water barrels) and the greenhouse covering it, that should be enough to turn a -2C or -1C outdoor temperature into a 1-2C inside the greenhouse.
Growing potted figs would also be beneficial. Start waking them up using the indoor warmth, possibly have them under grow lights for a few weeks so that the new growth is not too leggy, then bring them outside (gradually hardening them off to the outdoor sun) in May, and bring them inside if you get any late freezes.
Hardy Chicago is a relatively fast maturing variety, but it's not THE fastest. Those seem to be LSU Improved Celeste, Ronde de Bordeaux, Florea and Raasti, which should be a 1-3 weeks earlier.
If you're able to protect the previous year's growth from the cold (ex overwintering potted plants indoors - ideally still in a cool place where they can be dormant), you could get a good breba crop on it. Breba crops are typically about 4-7 weeks earlier than main crops (that are on new growth). Breba only grows on 2nd year wood though, not 3rd+.
Hardy Chicago generally produces little to no breba, and of poor quality, so I would get other varieties for that.
https://mountainfigs.net/mountain-figs/quality-breba-varieties/