The England international was dropped in favour of Costel Pantilimon after a series of mistakes earlier in the season.
He returned to the Premier League starting XI for the 4-2 win at Fulham on December 21 and has started every league game since then, although Pantilimon has still featured in the FA Cup and Capital One Cup.
Hart kept his latest clean sheet as City completed the double over the Cottagers with a 5-0 victory at the Etihad on Saturday but he accepts he has to constantly keep working to maintain his position.
Hart said: "I don't feel it's a case of that side of things being really solved.
"I'm still training hard - as I would be whether I was in or out of the team - and I'm always working to improve my game.
"I don't want it to seem as though everything is fine or that it was part of my life that I will forget, because I'm still living it and I am trying to live in the moment more recently.
"The past is the past and we don't know what lies ahead in the future so it's better to live day by day.
"We are going to get some wonderful opportunities at this club - some I will be playing in, some that I won't - and I'm just trying to look at things the best I can."
The victory over Fulham lifted Pellegrini's side to third in the Premier League, six points behind leaders Chelsea with three games in hand and Hart is confident City have the belief and ability to achieve their aim.
He said: "The fans, players, and staff - everyone needs to pull each other through if we want to get over the line first.
"I do think we have the squad to win the league. We can't get too wrapped up in the away games we still have to play.
"We still have the chance to do something really special and do something this club hasn't done for 44 years by winning two trophies in one season.
"So we need to press on and try and create a bit of history."