The North Korean delegation includes Vice Foreign Minister Choe Son-hui, who provoked Trump with her belligerent statements into canceling the summit last week, and Choe Kang-il, who is in charge of US affairs.
The man had been making rounds near the Marina Bay Sands hotel and the Esplanade yesterday ahead of the summit on June 12, when the actual Kim Jong Un will meet with Trump - the first time a North Korean leader comes face to face with a sitting United States president.
The weekend talks were the latest twist in a week of diplomatic ups and downs over the prospects for an unprecedented US-North Korea summit, and the strongest sign yet that the two Koreas' leaders are trying to keep the meeting on track.
The third team on a separate but complementary track to the other discussions was a CIA team Pompeo set up a year ago when he headed the spy agency.
Hagin served as deputy chief of Staff to President George W. Bush from 2001 to 2008.
The delegations are led by Ambassador Sung Kim, a former chief USA nuclear envoy, and North Korea's Vice Foreign Minister Choe Son-hui.
The Korean leaders' second summit in a month saw bear hugs and broad smiles.
"Chairman Kim and I have agreed that the 12 June summit should be held successfully, and that our quest for the Korean Peninsula's denuclearization and a perpetual peace regime should not be halted", Moon said.
Kim may see the sit-down with Trump as necessary to easing pressure from crushing sanctions and to winning security assurances in a region surrounded by enemies.
The meeting was thrown into doubt after Mr Trump pulled out last week.
The leaders of North and South Korea had discussed a three-way summit on April 27 at their first meeting in Panmunjom. Analysts said Mr Moon - who seems to have built trust with Mr Kim through two summits in a month - can help to bridge the gap between the United States and North Korea and mediate if necessary.
Still Trump tweeted that he believes the summit can move forward, and Kim, through the state-run news service, expressed a "fixed will" on the talks happening.
"Excessive worries receded as efforts resumed for a summit between the United States and North Korea", Okasan Online Securities strategist Yoshihiro Ito said in a commentary, which added that uncertainty remains.
They agreed to have their top officials meet again June 1.
Kim Chang-son is Kim Jong Un's chief aide, responsible for scheduling, itineraries, protocol and the flow of reports, policy documents and communications to and from the leader. The North hasn't openly repeated those same demands after Kim's sudden outreach to Seoul and Washington.
Moon has insisted Kim can be persuaded to abandon his nuclear facilities, materials and bombs in a verifiable and irreversible way in exchange for credible security and economic guarantees. Mr. Trump announced last week he was ending them over North Korean hostilities but now says negotiations have picked back up and he spoke Sunday as if the summit was a "go".
But within 24 hours he reversed course, saying it could still go ahead after productive talks were held with North Korean officials. The outcome of a summit between two world leaders is usually negotiated before they ever sit down at a table.