Pompeo met with Kim on Sunday morning for around two hours of talks followed by a lunch in the North Korean capital, AFP reported.
But a summit of the leaders of North and South Korea last month has helped to rekindle the peace process, as has the apparent desire of both Kim and Trump to meet again for a second summit.
Stephen Biegun, new US nuclear envoy who was accompanying the secretary, said he offered on Sunday to meet his counterpart, Vice Foreign Minister Choe Son Hui, "as soon as possible" and they were in discussion over specific dates and location.
After the summit between the leaders of the two Koreas last month, Kim said he was prepared to permanently dismantle his country's main nuclear site at Yongbyon, but only if the United States took "corresponding steps" to build trust.
North Korea's state newspaper, Rodong Sinmun, featured eight photos of the meeting on its front page, including shots of Kim and Pompeo smiling and shaking hands, as well as some with Kim's sister Kim Yo Jong.
In Tokyo, Pompeo told Abe that the USA wanted to work closely on nuclear talks and that he would raise with the North Koreans the issue of Japanese abductees held for decades by the regime.
A U.S. official added that the visit was "better than the last time" but further efforts were needed.
"I hope your trip to North Korea and the upcoming second US-North Korea summit will provide a good opportunity for achieving irreversible, decisive progress in terms of denuclearisation and the peace process on the Korean peninsula".
Pompeo said he and Kim discussed denuclearisation steps to be taken by the North and the issue of USA government monitoring of those actions, which Washington sees as vital, as well as the measures the United States would conduct in return, Moon's office said.
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He offered no details and upon landing in South Korea briefed White House national security adviser John Bolton and chief of staff John Kelly on his trip, officials said. It is to be noted here that US President Donald Trump had met Kim in Singapore in June for the first-ever summit between the countries. "So this is, I think, a good outcome for all of us", Pompeo said.
Beyond pressure from the president, Pompeo and his team are also facing growing pressure from China, Russia and even South Korea to ease sanctions on North Korea.
"President Trump wants the second meeting, and I think he can nearly spin anything coming out of North Korea to make the second summit happen anyway", said Terry.
In Tokyo on Saturday, Pompeo promised he would bring up Japan's concerns about North Korea's alleged abductions of Japanese citizens and its nuclear ambitions. "And we had a very successful morning so thank you and I am looking forward to our time here at lunch as well".
Pompeo is expected to meet Moon after his visit to North Korea on Sunday. But Trump called off the trip a day later, saying North Korea hadn't made sufficient progress toward denuclearization.
Pompeo plans to visit Beijing on Monday as he concludes his four-nation visit to the region that also included Japan.
In a sign of the modest objectives of the trip, Pompeo told a reporter en route to Japan that one of his goals would be to "make sure that we understand what each side is truly trying to achieve", a comment that signaled the two countries were still a long way off from detailed negotiations on verified disarmament.
He then planned to return in late August, only for Trump to cancel the trip at the last minute as it became apparent that the two sides remained far apart on their approach to the negotiations.
After a previous visit to Pyongyang in July, Pompeo had said the two foes made progress on key issues - but within hours of his departure the North condemned "gangster-like" demands from the United States, raising questions over how much the two sides really saw eye to eye.
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