MLI TRACON Controller reported an aircraft going into an MVA at a lower altitude than the MVA.

Narrative

VMC conditions. I had just received a position relief briefing from the previous controller on position. There were several VFR aircraft on frequency maneuvering; several VFR aircraft not on frequency; IFR over flights; IFR satellite departure; and 2 IFR aircraft doing practice instrument approaches at MLI. During the briefing; we received a verbal 'rolling call' on Aircraft X (one of the aircraft doing the practice approach) executing the published miss.The publish missed approach altitude is 2300 ft. They held for a moment; then requested radar vectors for another ILS [Runway] 27 approach. I assigned a 090 heading for a downwind vector. During this; I had PIA (to the South-East) and ZAU Sect 63 (DBQ to the North) calling simultaneously for 2 separate VFR point-outs on aircraft doing photo work on the northern and the other on the southern edge of my airspace. I also had a verbal call from the MLI Tower that the aircraft that just departed; which I received a non-verbal 'rolling call'; was the that aircraft because it did not auto-acquire (per our SOP). After all of the coordination; Aircraft X flew into an obstruction tower MVA of 2800 ft. (The highest obstruction is depicted on the chart as 1849 ft. MSL). The low altitude alert sounded and I climbed Aircraft X to 2800 ft. They continued the approach with no further issue.The MLI SOP Sec 3; 6-26; H states that Local Control advise the radar position responsible for departures if aircraft does not acquire on takeoff. Also; MLI SOP Sec 3; 7-32; A states aircraft exciting a missed approach to climb to 3000 ft.; but section B has them executing the published miss; which is currently 2300 ft. for all Instrument Approaches into MLI. My expectation bias was that Aircraft X was at 3000 ft. because published miss approaches are a lot less common. That combined with the multiple coordination at the same time; the assigned altitude on 2300 ft. was missed.I have suggested; several times in the past; that Sec 3; 6-26; H needs to change because this is covered in the 7110.65; 5-3-2; a. I was told that it was in there because of an incident that occurred here in the 1990's when an aircraft didn't acquire. We had ARTS IIe and CENRAP at that time. We now have STARS and ADS-B. Also; I would like to see a change to MLI SOP Sec 3; 7-32; A; that all missed approaches: planned; unplanned and published; climb to 3000 ft. for consistency.