South Side Park was the home of the Chicago American Giants and was located about 4-blocks south of the old Comiskey Park, if my hx lesson is correct, and continued to serve as the Chicago Negro League Team home at least into the 40s, am I right? South Side Park was the 1st home of the Chicago White Sox, (after they moved to Chicago from Sioux City where they were the Cornhuskers to St. Paul to Chicago).
That was actually South Side Park III (which looked awesome to me, at least the photos I remember seeing). You are right, it was a bit further south (39th St. which I think is Pershing Ave now I believe), but west of the Dan Ryan. The Negro League Team did play there once the AL White Sox moved to Comiskey in 1910, until 1940, the park burned down on Christmas Day I remember reading.
South Side Park (I) was really short lived, Chicago had a brief Union League team, that didn't even play a full season there. I think the team left town on a road trip and never returned in the summer. I think they reorganized that team in Pittsburgh or Philly (guessing 1884 which was the league's only season.) I think that park was a bit further east of the present day Dan Ryan Expressway.
I think it was both South Side II and III which owners of the clubs were trying to build the park closer to Ohio Street downtown, but had fierce opposition from NIMBYs and the
hoi polloi of Chicago. Street car lines were also big motivating factors for where to build.
The NL Chicago White Stockings (later the Cubs) eventually moved out to West Side Park (II) in 1893 I believe, which is where UIC-Med School is today. Some great stories involving that park in Crazy '08, one of my favorite baseball books. They played there before moving to Weegman (Wrigley) in 1916. It was actually the Federal League Chicago club that first played at Weegman the first two seasons.