The lion, called Sylvester, crawled under an electric fence after heavy rains dislodged earth, according to a Karoo National Park spokeswoman.
The park authorities intend to use a helicopter to find the lion.
He was fitted with a tracking collar after he was on the loose for three weeks last year.
More on this and other African news stories
The park authorities said on Wednesday that they were still considering five options, only one of which was to kill the lion.
Other options were to bring the lion back to the park and improve the fencing.
They said the previous decision to kill Sylvester had been made prematurely.
On Tuesday night South Africa National Parks tweeted that it was a difficult decision to put down the lion once they find him.
The hashtag #SaveSylvester has been trending in South Africa since Wednesday morning.
Last June, Sylvester went on a sheep-killing spree, wandering 300km (180 miles) before he was found taking a nap by rangers and airlifted from the Nuweveld Mountains in Western Cape.
On his current escape, he has already travelled at least 20km (12 miles) and killed a cow, according to AFP news agency.
Meanwhile in Kenya, another lion was shot dead by a ranger on Wednesday after he escaped from a park in Nairobi.
Several lions have escaped from the park in recent weeks.

Optimism fell for the third consecutive quarter according to the CBI employers group.
It is the sector's longest period of falling sentiment since 2009 - in the midst of the financial crisis.
The survey of 115 companies found low interest rates and potential restricted EU market access were seen as risks.
But while 28% of the respondents were gloomier, 15% were more optimistic.
And almost 40% of the firms surveyed reported healthy profits in the last quarter.
Rain Newton-Smith, the CBI's chief economist, blamed the dip in sentiment on the "uncertainty caused by Brexit to low interest rates, technological change and strong competition".
She called on the chancellor Philip Hammond to "set a clear direction for growth and prosperity" in his Autumn Statement in November.
"With firms voicing strong concerns about the impact of Brexit, especially the risks to the wider economy in the years ahead, the government must allay their unease with clear plans for negotiations to leave the EU," Ms Newton-Smith added.
The CBI campaigned for Britain to remain in the EU ahead of June's referendum.
Meanwhile, a separate report by the manufacturers' organisation EEF found that the UK's industrial sector had continued to strengthen in 2016, and British firms had created more jobs than equivalent companies in other developed counties, with the exception of the US.
Manufacturing only accounts for around 10% of the UK economy, but Lee Hopley, EEF's chief economist, said the sector's success was vital to the country's overall growth.
"This is especially important in a post-Brexit world," Mr Hopley added, "where we have to look at possibly all new avenues of generating growth and investment".
Animal rights campaigners are urging a South African game park to reverse its decision to put down a lion who escaped on Sunday.
[NEXT_CONCEPT]
UK financial services firms are becoming more pessimistic about their prospects in the wake of the Brexit vote, an industry survey suggests.