Red mentioned installing carpets when he was 15 at Lake Charlevoix (Season 2, Episode 3 “Dr. James Covington”) and working the Emmet County fair (Season 5, Episode 1 “Smokey Putnum”).
Assuming the real Reddington did not visit Michigan and Red is from there, Liz should be able to cross-reference Michigan’s records to find him.
That is if Red was telling the truth about those jobs.