The party whipped its Stormont MLAs on Monday to vote against a bill to outlaw the practice.
Party President Mary Lou McDonald had previously indicated she supported tightening such legislation.
On Tuesday, the party said its position was that “that regulation, not a ban, is the best approach”.
Alliance’s John Blair, who proposed the bill, had called on the party to clarify its position.
The North of Ireland is the only part of the UK where hunting wild animals with dogs is still permitted.
Monday’s vote in the assembly saw the legislation rejected by 45 votes to 38, with Sinn Féin, some DUP MLAs and the TUV leader Jim Allister voting against it.
Both the SDLP and UUP said they did not whip their MLAs to support the hunting bill as it was clear they were all supporting the party position which was to support the bill.
* All Alliance, SDLP, UUP, Green and People Before Profit MLAs who were able to vote supported the bill
* All Sinn Féin MLAs (25) who were able to vote, along with 19 of the 26 DUP MLAs voted against together with Jim Allister
Sinn Féin agriculture spokesperson Declan McAleer said the party opposed “the unnecessary infliction of cruelty to animals”.
“Our party position – north and south – is that regulation, not a ban, is the best approach,” he added.
“However Sinn Féin believes the Private Members Bill was unworkable, flawed and rushed and in the time we have left in this mandate amending it sufficiently was not possible.”
Earlier, John Blair had rejected that argument and told BBC’s Talkback programme he was “not buying it”.
“It adds to the puzzles already presented by Sinn Féin in relation to their position on this matter that they should seek, with some members of DUP and Jim Allister, to block a bill of this kind which public duty expects in the interest of animal welfare and then use reasons like that as justification to blocking progress,” he said.
“They know full well that the position on stage two of a bill is to discuss the principles of a bill and process. Timings and everything else that’s been cited by Sinn Féin would be worked out at committee stage and at further stages through the process.
“The committee chair, Declan McAleer, stated yesterday there wasn’t enough time for the committee to do this – but he must know full well that the committee he chairs looked at this issue last Thursday and agreed in principle that the committee would not need additional time for the committee stage of the bill.”
Animal welfare charity, the USPCA, had backed the bill.
Its chief executive Brendan Mullan was among those to criticise the result.
Speaking on Tuesday, Mr Blair accused Sinn Féin of a “cynical move” in rejecting the legislation at the second stage.
“Sinn Féin tried using various excuses to turn this into a debate about assembly time, but the second stage is about considering the principles of a bill.
“There was no reason at all in relation to process why this bill could not progress.”
Sinn Féin has yet to clarity the issue.
Mr Blair also said he hoped to reintroduce the legislation as soon as possible in the next assembly mandate, after the election in May 2022.
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