If it will be a Wednesday or Friday,
Club 81 Too would be my suggestion. About as Hegewischian as it gets. Keep in mind they only serve lunch from 11 to 2 (dinner starts at 4 on Wed, 3 on Fri).
I haven’t yet visited
M&M Windjammer but it certainly sounds worth considering.
A little farther north (not exactly Hegewisch) you could try
Small World Inn (3525 E 106th; scroll down in the post on Las Islas Marias, unfortunately no longer in business), a real old-time Serbian bar. Sad to report, Small World recently put up a For Sale sign so you’d be wise to call ahead (773-721-2727).
Again not quite Hegewisch, T’s Tap (9801 S Ewing) is another classic South Chicago bar where you can get good burgers etc.
I just saw Erik’s suggestions which are good ones. Be aware Casino has closed (and the promised Mexican place never opened). Also Steve’s Lounge (13200 S Baltimore), a great old bar, only serves food Friday evening (I think).