The Scot, who played Junior Wimbledon in his teens, next faces French 21-year-old Luca Van Assche.
Van Assche, who has a career high of 63 but is currently 186, defeated Juan Manuel Cerundolo, the world number 107 who was seeded three for the qualifying tournament, 7-6 6-2.
Before beating 31-year-old Broady, who has slipped to 632 in the rankings, Stewart had defeated another Englishman, Ben Jones, ranked 620, 6-4 6-3.
Playing mainly on the third-tier ITF Futures circuit, Stewart warmed up for Wimbledon qualifying with a 7-6 6-2 defeat by United States’ world 191 Murphy Cassone in an attempt to qualify for the second-tier Challenger event in Ilkley.
Should Stewart defy his ranking and oust Van Assche, he will need to then overcome either Swiss world 374 Leandro Riedi or Argentine 24th seed Federico Agustin Gomez, currently 134, to claim one of the 16 qualifying places for the main draw.
Wimbledon begins on 30 June, with Scotland’s Jacob Fearnley and Cameron Norrie already in the main draw.