Stanley remain bottom of League Two after suffering a home defeat by Cheltenham on Saturday afternoon.
The Reds were facing a side who were also looking for their first league win, just as they were at Bury last week, but again they were unable to take advantage.
Cheltenham midfielder Jason Taylor scored the game's only goal in the 71st minute to leave Accy with one point from their opening four league games.
The Robins had gone closest during the early stages with Taylor volleying wide from the edge of the box.
Having reverted to a 4-4-2 formation, Stanley then took charge in midfield but the best chances that came in the remainder of a quiet first half saw left-back Michael Liddle and Kal Naismith, switched from wide-right to a centre-forward role, firing wide from 25 yards.
The match livened up after half-time with the visitors, who reached the play-offs the last two seasons, getting on top.
Steven Gillespie and Sam Deering fired wide while Taylor's deflected strike was easy for Ian Dunbavin.
The Reds replied with Wilson dragging a 20-yard effort well wide but Cheltenham then broke the deadlock from a free-kick that should never have been given.
The referee saw a textbook block tackle by Peter Murphy, covering for the injured Nicky Hunt at right-back, as a foul, and after the Accy defence failed to deal with Matt Richards' free-kick, Taylor forced the loose ball past Dunbavin from close range.
But Stanley finished with 10 men as Wilson was forced off with injury after all three substitutions had been made and the nearest either side went to scoring late on was when Deering poked wide on the break.
Stanley : Dunbavin 6, Murphy 6, Aldred 7, Atkinson 7, Liddle 6, Mingoia 6 (Gray 66, 6), Richardson 5 (Hatfield 82), Joyce 6, L. Wilson 7, Naismith 7, McCartan 6 (Carver 74, 6). Subs: Clark, Windass, B. Wilson.