Despite edging into the top six of the SPL in their final match before the split, a top-four finish and the guaranteed Europa League slot that comes with it remains a possibility following weekend defeats for Motherwell and Hibernian.
Hearts will move one point behind Motherwell in the Clydesdale Bank Premier League if they beat the Lanarkshire side at Tynecastle on Saturday.
They also have Hibernian in their sights with an Easter Road derby followed by Tynecastle clashes against Dundee United and Celtic. Even a fifth-placed finish could secure a European place, depending on Dundee United's league finish and Scottish Cup final result, and Jefferies wants Hearts fans to recognise the importance of their run-in.
The manager told Hearts News: "Saturday is the first of three home games for us before the season ends and if we want to finish as high up the league as we can, then we have to make the most of that advantage by showing desire and commitment.
"The support the fans gave the team against Kilmarnock was superb and they have an important role to play in the next few games. We now don't have to leave Edinburgh before the season finishes so there isn't too much of a problem in terms of travel for the supporters."