Worcester recorded a tense 13-6 league victory over London Welsh last time out to maintain the club's impressive home form that sees the team unbeaten at Sixways since the opening day of the campaign in all competitions.
Hill knows away day blues must now also be banished in 2013 if the team are to become serious contenders for a top six finish.
However, he remains confident of more success in the New Year and said: "Three or four games ago I think we dispelled any thoughts about worrying about relegation.
"I have had the impression, from talking to supporters and staff working here at Sixways, that after three or four games they saw the way the team was developing and playing and, maybe for the first time, said to themselves 'We are not going to go down'.
"We have turned our attention to getting into the top six. We knew this seven match run - that started against Saracens and finishes with Leicester - would dictate whether we could get into the quarter finals of the Amlin Challenge Cup and the top six of the Aviva Premiership.
"With wins over Saracens and London Welsh, and a losing bonus point away at Quins, it is a realistic option. But we have two huge games to go to really push home the advantage.
"If we can beat Sale and maintain our excellent home record against Leicester then we will be serious contenders."
Hill has also revealed how his squad have this year had to put festive celebrations to one side to concentrate on pushing up the league table and big games ahead.
He added: "What have I told the players about this time of year? Well, they are not allowed to eat any Heroes or Celebrations - we have banned tins of chocolates in the office because it is very easy to pick out two or three!
"Seriously though, they were in up until Christmas Eve, had Christmas Day off and then they were back in early on Boxing Day morning.
"They have a small window to spend time with family and friends. Unfortunately that is what we have got to do because we are taking all these games very seriously.
"Last year we played Leicester here and I know we had too much time off for Christmas. Leicester treated that one very seriously and worked through Christmas and it paid off because they beat us here. Now we are doing exactly the same."
Watch the Warriors in New Year action against Leicester Tigers on Friday, January 4 at Sixways! Call the Warriors Box Office during opening hours on 01905 454183 or click HERE to buy online.