Indigent rhymes with:
diligent, exigent, agent, astringent, bagent, cogent, contingent, convergent, detergent, diligent, divergent, emergent, exigent, indulgent, insurgent, intelligent, intransigent, largent, legent, migent, negligent, neugent, nugent, pageant, pungent, reagent, regent, resurgent, sargeant, sargent, sergeant, sergent, stringent, tangent, urgent, vigeant
Indigent sounds like:
indecent, indignant, indignity, inducement, integument