John McDonnell, Diane Abbott and Richard Burgon all criticised the party’s campaign.
However, Sir Keir vowed to pick up the pace of change away from the Corbynite policies and leadership which led the party to its disastrous 2019 election defeat.
Shadow Communities Secretary Steve Reed told BBC Radio 4’s Today programme: “We have a breach of trust between too much of the British electorate and the Labour Party, and we haven’t done enough to repair it over the past year.”
Former Blairite Cabinet minister Lord Mandelson also stressed that the party needed to fight the next election on the Centre ground.
“The last 11 General Elections read ‘lose, lose, lose, lose, Blair, Blair, Blair, lose, lose, lose, lose’,” he said.
“We need for once in this party to learn the lessons of those victories as well as those defeats.”
Leftwingers, though, were quick to criticise Sir Keir.
Former shadow Chancellor Mr McDonnell told BBC Radio 4’s Today programme: “In any election you need to have an argument..you have to put up an argument..the Labour Party went into this election campaign..it was like having an argument without putting an argument...or a campaign without putting a campaign based upon what you wanted to do or what sort of society you wanted to build, or the policies you want to advocate.
“We should never, ever do that again.”
He added: “You can’t treat working people like lobby fodder, you have got to really go along and work in the community over years, talking through the policies that you want, listening to them.”
However, he said Sir Keir should be given a chance to show if his leadership could succeed, adding that there had been some “carry over” of Brexit in the Hartlepool result.
Former shadow home secretary Ms Abbott tweeted: “Crushing defeat for Labour in Hartlepool. Not possible to blame Jeremy Corbyn for this result.”
Former shadow justice secretary Richard Burgon messaged: “Incredibly disappointing defeat in Hartlepool. We are going backwards in areas we need to be winning.”
The Labour Leftwinger added: “Labour’s leadership needs to urgently change direction. It should start by championing the popular policies in our recent manifestos - backed by a large majority of voters.”
The Momentum movement, which backed Mr Corbyn, messaged: “This result is a disaster. In 2017, we won over 50% of the vote in Hartlepool.
“A transformative socialist message has won in Hartlepool before, and it would have won again.”
However, Labour centrists and Rightwingers rejected the calls for the party to return to the Corbynite era which led to the historic 2019 defeat.
Shadow Communities Secretary Steve Reed described Labour’s defeat in the Hartlepool by-election as “shattering”.
Speaking on the BBC’s Today programme, he said: “It tells us that the pace of change in the Labour Party has not been fast enough...
Lord Mandelson, a key figure in the Blairite administration, tore into Leftwingers calling for a shift back to policies championed by Mr Corbyn.
“What I would say is this and remind the party we have not won a General Election in 16 years.”
He believes Labour’s dismal performance in Thursday’s elections can largely be put down to Covid and Corbyn, with many voters still having not forgiven the party for choosing him as leader.
Phil Wilson, who in 2019 lost Sedgefield, Tony Blair’s former constituency, told @LBC Radio: “The overhang from the Corbyn-era is still there.
“The world view that had no affinity with country. My view is that if you have no belief in your country, why do you want to govern it?”