On Friday, in the biggest such demonstrations since communism fell in 1989, some 50,000 anti-government protesters rallied in Bratislava.
Slovakia's interior minister will resign to prevent the collapse of Prime Minister Robert Fico's government, which is trying to fend off calls for early elections following the murder of an investigative journalist.
After resisting calls to resign and riding out several previous scandals, interior minister Robert Kalinak resigned on Monday, depriving Mr Fico of one of his closest allies and a co-founder of their populist Smer party.
"We think that only early elections would solve this situation", said Bela Bugar, chairman of Most-Hid on Monday evening after an eight-hour internal party discussion.
Slovak investigative journalist Jan Kuciak and his fiancee were killed in their home last month.
The junior nationalist coalition party SNS said on Monday that it could accept either early elections or a cabinet reshuffle.
President Andrej Kiska also has proposed early elections or substantial changes in the government as a way out of the crisis.
Bugar said Fico rejected the Most-Hid party's proposal for a snap vote.
The coalition, which also includes the Slovak National Party, has a slim majority of 78 deputies in 150-seat parliament, including 14 lawmakers for Most.
"It's important for stability to be maintained, therefore I chose to resign the post of deputy prime minister and interior minister", Mr Kalinak said.
Kalinak is Fico's key ally in Smer-Social Democracy, which has previously been alleged to have links to corruption.
The Most-Hid party in Slovakia has threatened on Monday (12 March) to withdraw its support from the coalition government, if the country does not hold early elections.
An worldwide team of journalists has been formed to complete Kuciak's story.