Mr Abe’s body was taken to his home in Tokyo after he was shot in the western city of Nara during a rally speech, and later died in hospital.
Nara prefectural police chief Tomoaki Onizuka said Abe’s assassination was his “greatest regret” in a 27-year career.
“I cannot deny there were problems with our security,” Onizuka said. “Whether it was a setup, emergency response, or ability of individuals, we still have to find out. Overall, there was a problem and we will review it from every perspective.”
Mr Abe’s death came ahead of the country’s Upper House elections on Sunday.
Japanese Prime Minister, Fumio Kishida’s ruling Liberal Democratic Party (LDP) was projected to keep a majority in the upper house with its junior coalition partner in Sunday’s election, public broadcaster NHK reported.
The LDP and the Komeito party are expected to win between 69 to 83 seats out of the 125 contested in Sunday’s vote, according to NHK’s exit polls.
There was an increased police presence for Mr Kishida at a campaign event in a city southwest of Tokyo and a metal detection scanner was installed at the venue, an unusual security measure in Japan.
The attacker who killed Mr Abe told investigators he believed rumors that the former prime minister was connected to an organisation that he resents, police said.
Japanese media reported that the man had developed hatred toward a religious group that his mother was obsessed about and that caused his family financial problems. The reports did not specify the group.
Police on Saturday said autopsy results showed that a bullet that entered Abe’s upper left arm damaged arteries beneath both collar bones, causing fatal bleeding.
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Japan is particularly known for its strict gun laws. With a population of 125 million, it had only 10 gun-related criminal cases last year, eight of then gang-related.
