The Tynecastle outfit have slumped to second bottom of the standings after losing their last five Clydesdale Bank Premier League games.
First-team coach Gary Locke was in charge for the most recent of those matches - Saturday's 2-1 defeat to Motherwell - after boss John McGlynn was sacked last week.
Hearts now have Tuesday's home encounter with St Johnstone and Sunday's Edinburgh derby clash with Hibernian at Easter Road - both in the SPL - before meeting St Mirren for the League Cup decider on March 17.
Player aspirations to succeed at Hampden Park will be fading after last week's developments followed various off-field financial problems and a string of recent departures including those of first-team regulars Ryan McGowan and Andy Driver.
Stevenson claims even landing the cup - and a first winner's medal of his career - might not be enough to restore the club's battered pride.
He said: "We need to start picking things up now because we have got three massive games coming up and us sitting 11th in the table is nowhere near good enough.
"It's been a season we'd all like to forget. Obviously we have got the cup final coming up, but regardless of that, we are second bottom and the games are running out.
"Would a cup final win change that? I don't really know. I've never been in a situation to win something.
"Maybe a week on Sunday if we win the cup, I can possibly sit there and say it's been a great season, because I've won my first medal. It's been a long time for me personally to have the chance of winning anything.
"But since I came here, we have always been pushing up the top end of the table. So for us to be sitting here down the bottom is something I can't stop thinking about.
"Would it take the shine off a cup win then? Possibly yes. Your league campaign is your bread and butter and as much as you go on about a cup run, the fact your manager has lost his job and it's getting harder and harder to make the top six is a major disappointment."
McGlynn was relieved of his duties the morning after last Wednesday's 2-0 league defeat to their cup-final opponents in Paisley.
It was the 15th defeat from his 35-game spell in charge of Hearts after he quit Raith Rovers to rejoin a club where he had previously headed the youth development programme.
But Stevenson insisted McGlynn could not have done much more to try to make a success of his tenure and it was for that reason that he personally called the dismissed manager to offer his sympathies.
He said: "I spoke to John on Friday. I have nothing but admiration for him - I can't speak highly enough of the guy.
"Day in, day out, he worked his socks off for the club and was always there for his players. It would be good to see him back as soon as possible, because the game needs guys like that.
"I imagine he would have been low at that point when we spoke. It's a club he supports and it was his dream job. For him to only have eight months at it would probably have been hard for him to take.
"But he's a football man and he knows the results haven't been good enough. It's hard as well because the players have got to take a look at themselves too. There is only so much the manager can do. Ultimately he has paid the price for us not doing well enough.
"It wasn't an easy phone call, but sometimes you have to make it.
"Sometimes when you are one of the older, experienced ones in the team, you have got to think about that. At the end of the day, John has lost his job and it's a horrible situation. In my career I've lost too many managers.
"Thursday was a sad day for me, because John was one of the nicest men I've met in football."
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