Smidgen rhymes with:
bijan, pigeon, pridgen, religion, allergan, allergen, amerongen, antigen, belgian, bijan, bludgeon, burgeon, cajun, carcinogen, collegian, contagion, dajun, detjen, dodgen, dudgeon, dungeon, engine, estrogen, evjen, georgian, gudgeon, halogen, hodgen, hydrogen, imagine, karajan, legion, margin, nadjen, neurosurgeon, nitrogen, norwegian, oncogen, origin, oxygen, pathogen, petitjean, pigeon, pridgen, rathjen, region, religion, sturgeon, sundararajan, surgeon, tejan, tietjen, trajan, trojan, trudgen, virgen
Smidgen sounds like:
semitism, smithson, smithsonian, somatogen, soundscan, swendsen