Dundee 2 Rangers 1
The Ibrox side are now 6 points adrift of Aberdeen in the race for second place in the Premiership.
Rangers' need for fresh leadership became even more pressing as they suffered a 2-1 defeat to Dundee at Dens Park.
Caretaker boss Graeme Murty was in charge for the second match running following Mark Warburton's exit.
But he was left looking forlorn as goals from Mark O'Hara and Kevin Holt gave the Taysiders their first home win over the Ibrox side in 25 years.
Joe Garner gave Rangers hope after the break but the damage had already been done.
Over a week on from Warburton's exit, the Light Blues board has yet to give any hints as to who will take over on a permanent basis.
That matter, however, will have to be addressed urgently with their side now six points adrift of Aberdeen in the race for second place in the Ladbrokes Premiership.
Paul Hartley's Dundee side, meanwhile, climb back into the top six.
After scrapping past Scottish Cup opponents Morton last week, Murty hoped to inject some life into his team by replacing the injured Lee Wallace, ineffective Michael O'Halloran and out-of-sorts Philippe Senderos with Lee Hodson, Garner and Rob Kiernan.
But it was Dundee who came out flying.
They went after the Light Blues straight from kick-off. Craig Wighton ghosted past Kiernan but O'Hara put his header wide when he should have scored.
That was a warning but Rangers did not heed it.
They fell behind after 13 minutes as Dundee were allowed to counter swiftly from the visitors' own corner.
Hartley's team worked the ball to Henrik Ojamma in acres of space down the left, with the Estonian cutting back for O'Hara who was again left free in the box. All he had to do was hold his run at the penalty spot before stroking in off the post.
The large travelling support did their best to raise a response but Rangers looked simply lifeless.
The best they could muster during the opening half were a couple of tame Kenny Miller strikes and a Garner header which drifted wide.
But worse was to come for Murty's men four minutes before the break. Substitute Danny Wilson - on for Clint Hill, who trudged off after a clash of heads - tripped O'Hara to give the Dark Blues a free-kick on the corner of the box.
Rangers number one Wes Foderingham lined up his wall to cover the near post but was still embarrassed by Holt's whipped shot. Wrong-footed, the Englishman could only push the ball into his net.
Rangers did respond after the break, with Dundee keeper Scott Bain tipping James Tavernier's free-kick over the bar.
And they got their lifeline after 62 minutes when Garner took aim from 25 yards out, sending a swerving shot away from Bain into the bottom corner.
The fight was suddenly restored to the Ibrox side's play.
They came close to levelling when Miller's strike was deflected off James Vincent onto the post.
But substitute Harry Forrester failed to convert a golden opportunity with 10 minutes remaining that would have squared it, while Foderingham had to save his side further misery late on as Wighton raced through.