Abbott, assistant Graham Kavanagh and first-team coach Tony Caig all put pen to paper on new one-year deals last week and will now set out to win back those stay-away supporters.
"We had some very poor results last season," Abbott told the club's official website. "That affected our crowd and the knock-on from that was to the financial side of things at the club.
"We had to deal with that and I think it was very important that we got through it and kept our place in what is a very tough division."
The Cumbrians flirted with relegation before an unbeaten six-game run in March and April lifted them clear of trouble. They finished seven points clear of the bottom four.
"Fair play to the players because they got the results when we needed them," Abbott added.
"That put is in the position we're now where we can look at what happened and start the job of rebuilding so that we can see more of the success we all want.
"It's a difficult job, but we're ready and prepared to take it on."