McCoist and his dejected players were subjected to abuse from furious Gers supporters after the final whistle at Easter Road last night.
They passed up several excellent scoring chances in the capital before crashing to a 1-0 defeat to SPFL Championship club Raith Rovers.
However, the Light Blues boss insisted he was focused on the future - and the William Hill Scottish Cup semi-final with Dundee United at Ibrox on Saturday.
Despite the loss to Raith and despite the injury sustained by left-back Lee Wallace, he is still confident his side can beat their SPFL Premiership rivals.
He said: "I don't fear for my future at all, no. There's nobody will need to tell me anything (about when to go), that's for sure.
"I've been here long enough as a player, coach and manager to know there are a lot of unhappy people - none more so than ourselves, the players and staff.
"So as down and as low as we are, we can't stay down and low, because we've got a massive game coming up next week.
"We'll regroup, we'll see how the injuries are in the next 24 hours, and we'll start to prepare for a massive game, the semi-final."
McCoist added: "United on their day are more than a handful for any team in the country, and we will have to play a lot better than that.
"We'll certainly have to take our chances. But I definitely do believe that we can overcome them."
McCoist, though, admitted the pain he felt at the cup final loss to Grant Murray's side was as bad as any he had experienced in his playing or coaching career.
He said: "I've had some big disappointments, but that's up there with all the major disappointments that I've had.
"But it's not just about me. It's about the team and the club and the players, all of us together. So it's a big disappointment and we need to take it on the chin.
"We don't have time to feel sorry for ourselves. That's the one thing we can't do. We've got a big game to look forward to on Saturday."
McCoist refused to accept that Rangers must win the Scottish Cup last four tie with United in order to safeguard his future as manager.
He said: "I'm not sure it's a must-win. We're playing one of the better teams in the country, in a cup semi-final for which they'll be favourites.
"We want to win it, that goes without saying, whether it's a must-win I'm not so sure."
McCoist admitted he was disappointed with the quality of the Rangers defending that lead to John Baird netting a dramatic late winner for Raith.
He said: "There was a slackness on occasions about us defensively and it came back to bite us.
"I think that's the first time we haven't scored this season in a game. It's the second defeat, albeit both of them have been in extra-time, but it still hurts.
"I thought the manner in which we lost the goal was really, really poor from our point defensively.
"We had great chances -there was nothing really clear-cut, but I thought we should have stuck one of the headers away."