The Tynecastle outfit are now under the control of finance trouble-shooters from BDO as they battle their cash crisis.
Their insolvency woes last week triggered a Scottish Premier League points penalty that now has them favourites to be relegated from the top flight.
But full-back Hamill - who on Tuesday agreed to take a 50% wage cut - insists the club can overcome the 15-point and thrive in the SPL.
And he claims it will be a bigger achievement than the Gorgie side's 5-1 drubbing of their city rivals in last year's William Hill Scottish Cup final - or any of the club's trophy triumphs.
Hamill told Press Association Sport: "Can we overcome the 15-point penalty and beat relegation? Of course we can - otherwise I wouldn't have stayed on.
"What is the point of signing on again just to get relegated? I'm here to do well and I think we can move on from the 15 points fine and compete.
"If we can overcome that punishment it will be bigger than any Scottish Cup victory this club has ever had. It will be the biggest achievement in the club's history.
"We've still got the core of the team from last year here. Hopefully we can push on and put all the negatives behind us.
"Don't get me wrong - it is going to be difficult. But if we can get to 11th in the league, everyone will be very happy."
- Sports & Recreation
- Scottish Premier League