"The United Nations has tremendous potential and that potential is being met, slowly but surely", Trump told the drug meeting attended by UN chief Antonio Guterres.
The deal appeared created to stay within parameters that would be acceptable to the USA, which maintains some 28,500 troops in the South.
Bolton said the United States was seeking to ramp up pressure on Iran but not to overthrow the regime - an idea he supported before taking his job and reiterated recently by former NY mayor Rudy Giuliani, who is Trump's personal lawyer.
The president further stated the US and South Korea will continue their great economic partnership together.
Trade analysts say changes to the South Korea agreement were largely cosmetic, as the president limited the scope of renegotiations by declining to invoke U.S. trade law that would have required congressional approval of the final deal.
He said USA automobiles, pharmaceuticals and agricultural products will gain better access to Korean markets.
Moon says companies from both countries will be able to do business under more stable conditions.
It's also unclear how the military measures could affect the armistice agreement, which is enforced by the UNC under the command of Army Gen. Vincent Brooks, who also leads U.S. Forces Korea.
"As you know Kim Jong Un wrote a letter - a handsome letter - asking for a second meeting and we will be doing that", added Trump, who met with Kim in Singapore in June. "I am going to be meeting with Chairman Kim in the not-too-distant future".
One thing we know for sure from the summit between the leaders of South Korea and the USA,.is that there will be a redux of the Singapore summit, . hopefully pushing efforts to denuclearize the Korean Peninsula in the right direction.
Mr. Trump also hopes by Wednesday to persuade Japan to enter formal bilateral trade talks, part of a commercial diplomacy.
His national security adviser, John Bolton, said that Trump would stress United States defense of its sovereignty in his latest UN address on Tuesday.More news: Gunmen attack army parade in Iran leaving dozens dead
A senior Russian lawmaker who is under USA sanctions is pushing for the release of jailed Russian gun activist Maria Butina during a special visit to NY.
The nuclear threat also was on the agenda at Trump's first meeting in NY, a dinner with Japanese Prime Minister Shinzo Abe on Sunday night.
Slutsky is under US sanctions over Russia's annexation of Crimea, but won special permission to come to NY to attend the United Nations meetings.
Efforts to resume economic cooperation are limited by US -led sanctions linked to the North's banned nuclear weapons program.
Secretary of State Mike Pompeo (pahm-PAY'-oh) said Monday on the sidelines of the United Nations General Assembly, "We've been at this the other way an awfully long time and failed". We tried to do step for step.
The US president similarly described the trade deal with Seoul as a breakthrough and completely unlike an earlier deal, known as Korus, negotiated by the Obama administration. Trump said he will meet with Rosenstein on Thursday upon his return to Washington.
Foreign Minister Jean-Yves Le Drian says the U.N. General Assembly this week will test whether people like Trump are "playing together or playing lone horseman, whether they are seeking solutions or sticking to slogans". Le Drian suggested the ceremonies could remind Trump of the importance of global cooperation.
Mr. Trump, in his meeting with French President Emmanuel Macron, said the two have had great experiences, "on occasion not so good but 99 percent very good".
The UN's number one financial backer, the United States has moved under Trump to cut funding to the world body, notably to peacekeeping missions that are key to the UN's goal of promoting peace and security.
Trump used his debut address to the UN General Assembly 12 months ago to threaten to "totally destroy" North Korea and belittled its leader as "rocket man", prompting Kim to respond by calling the USA president "mentally deranged".
He said Monday: "It was a different world".