Victory leaves them 10th but if they win Saturday’s final match at home to seventh-placed Chesterfield (12.30pm) then Halifax could claim a play-off place.
Just two points separate the teams from fifth to 10th, though Eastleigh, in ninth, have a game in hand at Altrincham tomorrow.
Danilo Orsi got one back for 10-man Maidenhead on Saturday but Halifax held on for a deserved win.
Halifax’s cause was given a boost when Ryan Upward was sent off for a second booking after a trip on King, subject of abuse from a section of the home crowd from the kick-off.
That dismissal entrenched the Shaymen as the more commanding side after they had grown in confidence from a quiet start.
Hyde capped an encouraging opening half when he glanced the outstanding Luke Summerfield’s corner across goal and into the far corner for Halifax’s first goal in three games.
Shortly after King’s effort was ruled out for offside, there was a big chance for Stephenson a few minutes after the interval from Kieran Green’s low cross, but Holden did well to spread himself and save the shot with his legs.
Halifax were the dominant side, as expected with their man advantage, but with the home crowd doing their best to rally the hosts, the comfort of a second goal was desirable.
It came when King stroked in a wonderful free-kick across goal from 25 yards after he had been fouled.
Top-scorer Orsi produced the hosts’ first attempt on target after an hour, but Sam Johnson reacted well to kick it away.
That was a warning, though, and moments later Orsi, who was announced as the club’s player of the year at half-time, produced a close-range finish from a long throw into the area.
Goalkeeper Holden produced an excellent save to keep out King’s rising drive after an incisive pass by Summerfield had played in Neill Byrne down the right.
There was a routine save for Johnson from Dan Sparkes’s curling shot before Woods’s volley was cleared at the other end,
Maidenhead: Holden, Massey, Wells, Parry, Sheckleford, Coley (Wiltshire 85), Ince, Sparkes, Barratt (Keetch 73), Upward, Orsi. Unused substitutes: W Smith, Twumasi.
FC Halifax Town: Johnson, Byrne, Clarke, Maher, Senior, King (Campbell 91), Summerfield, Woods, Green (Tear 83), Hyde, Stephenson. Unused substitutes: Davison-Hale, Spence, Bradbury.
Referee: T Reeves.