Saturday's 3-0 defeat at the hands of Dundee United saw Rangers dumped out of the Scottish Cup, having already exited the Ramsdens Cup and the Scottish Communities League Cup.
The Light Blues are 19 points clear at the top of the Third Division, meaning title success in the league is now just a matter of time.
But, despite appearing to have little left to contest, McCoist says there is no chance he, or the club's supporters, will allow their campaign to simply fizzle out.
He said: "We've got everything to play for in terms of our main objective at the start of the season, which was to get promotion from the league.
"The players will certainly know their responsibilities and duties are to go out and make sure it's not flat.
"Our fans have been nothing short of fantastic and I'm sure they will not allow any flat atmosphere within any of the grounds we go to.
"It's my job to make sure the players are up for it - and they will be."
McCoist had claimed beforehand that United would be favourites to progress from the fifth round of the cup.
But Rangers were left reeling when their hosts surged into the lead after just 16 seconds as Johnny Russell took advantage of poor defending.
Jon Daly doubled the advantage with a free header, before Russell completed his double in the second half.
A poor day for the Glasgow giants was compounded by the dismissals of Kal Naismith and Ian Black late on.
McCoist added: "It was disappointing and you would have to say the better team won.
"It was a disastrous start to the game. You work all week on your shape and it gets thrown to the wall after a minute.
"The second goal as well was really poor defensively.
"We came into it more in the second half for 25 minutes and then, again, we gave away a really poor third goal."
The win was United's first at home since August and allowed their new manager Jackie McNamara to enjoy a positive start.
Russell said: "I was delighted to get on the scoresheet myself but the most important thing was getting into the next round of the cup.
"It was a big game for Rangers but it was a massive game for us as well.
"They are still massive games when you play against Rangers and that victory feels just as good as it would if they were still in the SPL."
Sheffield United reportedly failed with a £400,000 bid for the striker on transfer deadline day.
Russell said: "I didn't have a clue about it. It wasn't what the club wanted and they rejected it.
"I'm really excited to work with the new manager.
"He's not had a lot of time but it has given us all a boost with the new gaffer coming in and hopefully we can continue that for the rest of the season."