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 night'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."