Simmonds, who was previously at Nuneaton Town, scored in the 78th minute of a 2-2 draw with Maidenhead United.
In the Conference North, Nuneaton Town drew 1-1 at Solihull Moors. Simon Johnson scored for Moors in the 36th minute but Armson equalised in the 46th minute.
In the Southern League Central Division, Rugby Town drew 0-0 at home with
Woodford United but there were defeats for Bedworth who lost 3-0 at Arlesley and Atherstone who led 1-0 through Weale but lost 2-1 at home to Leighton Town.
Stratford's Midland Alliance game at Studley was postponed.
In the Midland Combination Premier Division, Nuneaton Griff beat Racing Club Warwick 2-1. Racing opened the scoring through a 43 minute T.Smith goal but second half goals from Flavell and Piggon gave Griff the win.
Alveston's game against Southam was postponed as were the Coventry Charity Cup ties between Kenilworth Town and Massey Ferguson and Stockingford and Coventry Copsewood and Hawkesmill Sports and. Stockton.