It was unclear how many casualties the third blast caused or whether it had any connection to the earlier bombings.
The latest attack took place behind the Kabul bank in the Shar-e-Naw area of the city, according to a police officer at the scene. The police officer described one explosion carried out by one attacker inside a car.
Taliban terrorists claimed responsibility for the first two bomb blasts Monday near the Afghan Ministry of Defense in Kabul that killed two dozen people.
Saleem Rasooli, head of Kabul hospitals, told CNN at least 24 people had been killed in the attack, which also injured 91 others.
The afternoon attack included an initial bombing, followed by a suicide bomb attack, said Sediq Sediqi, spokesman for the Afghan Ministry of Information.
The first blast was a magnetic bomb and the following suicide attacker was wearing an Afghan national army uniform, said Fraidoon Obaidi, chief of the Kabul police Criminal Investigation Department.
Among the dead, which included both security forces and civilians, were the head of Kabul's second police district and his deputy, Obaidi said.
Taliban spokesman Zabiullah Mujahed tweeted that the Taliban had carried out the attack, the latest of several kidnappings and bombings blamed on the hardline religious terrorist group.
Afghan President Mohammad Ashraf Ghani conveyed his condolences to the victims' families and denounced the methods of terrorist groups.
"The enemies of Afghanistan have lost their ability to fight the security and defense forces of the country and thus attack highways, cities, mosques, schools and common people," Ghani said in a statement issued by his office.
The increased violence has heightened security fears in Kabul, Afghanistan's capital city.
Earlier this year, American and other diplomats were barred from traveling by road from Kabul's airport to their diplomatic missions a short distance away. Instead, they've been ferried by helicopter.
Many countries, including the United States and United Kingdom, have had longstanding travel advisories against all but essential travel to Afghanistan because of the security situation.
The Taliban has a long history there, dating back to the early 1990s when the country was still dominated by Moscow. Taliban fighters waged a civil war against the Russian-backed government, eventually taking control of Afghanistan in 1996. After Taliban leaders sheltered Osama bin Laden and other al Qaeda terrorists, its government was crushed by a US-led invasion in 2001.
Since then, Taliban fighters have been slowly building an insurgency against US-led forces and the US-supported Afghanistan government.
Afghan government forces engaged in days of heavy fighting with the Taliban in the key northern province of Kunduz, at one point losing control of the district.
Fifteen years of fighting against US-led forces has contributed to fractures inside the Taliban. The group has begun to include younger radicals and more criminal elements.
Experts describe Akhundzada as a religious scholar and a member of the Taliban's founding generation. He was close to founding Taliban leader Mullah Omar
, said Thomas Ruttig of the Afghanistan Analysts' Network.
A source in the Taliban reached through an intermediary told CNN last May that Akhundzada told commanders and shura members that there will be no peace talks.
The source said the appointment of Akuhnzada "will bring back the era of Mullah Omar" with "a simple life, loyalty, and terror on enemies."
Last year, Haseeb Sediqi, spokesman for Afghanistan's intelligence service, confirmed rumors of Omar's death,
saying he died in a hospital in Karachi, Pakistan, in April 2013.
Akuhnzada's deputy, Sirajuddin Haqqani, is known by the United States as the main facilitator for al Qaeda in Afghanistan.
Haqqani has a $10 million US State Department bounty on his head and heads an extensive family-based criminal network based primarily in North Waziristan, Pakistan. It held captured US soldier Bowe Bergdahl before he was repatriated in a prisoner swap.
Ruttig said Haqqani and another Akuhnzada deputy -- Mawlawi Mohammad Yaqoob -- represent a younger, more militant generation of the Taliban.