The 29-year-old started on the right of defence in the club’s 2-1 friendly win over Partick Thistle on Sunday.
Manager Robbie Neilson revealed on Saturday that those who didn’t start against Ayr United in the Championship fixture would be given minutes in the bounce game.
The match allowed him to experiment with the right-back position ahead of Smith’s absence.
The Northern Irishman will miss the league fixture with Queen of the South and Scottish Cup tie with Brora Rangers due to international commitments alongside Liam Boyce.
Neilson had been keen to bring in a right-back on loan following Jamie Brandon’s season-ending injury but no deal materialised.
White, therefore, was given the nod at right full-back with Cammy Logan, who is on loan at Cove Rangers, playing on the left against Thistle.
There were also starts for Peter Haring, Chrsitophe Berra, Jamie Walker and Gervane Kastaneer.
Thistle took the lead through Shea Gordon after being picked out by former Hibs striker Brian Graham.
Hearts, wearing last season’s kit, were level before half-time and it was Euan Henderson with a confident low finish, following on from his brace in a recent friendly with Motherwell.
Assistant boss Lee McCulloch has spoken previously of the forward's goal scoring ability.
“Fantastic finisher with both feet," he said. “I think he just needs a little bit more confidence and belief in himself.”
The winning goal was scored by Berra before the hour mark.
Hearts team: Stewart, Logan, White, Darge, Haring, Berra, Frear, McGill, Henderson, Walker, Kastaneer. Subs: Stone, Hamilton, Smith, Cochrane, Flatman, Denholm