The UN is pushing for peace talks by the end of the year, and Sweden said it is ready to host them.
During the past few days, the government forces made significant progress despite a large number of landmines and booby-trapped buildings, in addition to professional Houthi snipers who took positions on buildings inside residential neighborhoods, local military sources said.
Britain, the United States and France have all called for a cessation to the hostilities.
He urged all parties to relaunch peace talks, which were abandoned in September.
During a visit to Riyadh and Abu Dhabi on Monday, Hunt met Saudi Arabia's King Salman and Crown Prince Mohammed bin Salman, as well as leaders from the United Arab Emirates and Yemen.
But residents in Hodeida, whose port is vital for food imports and humanitarian aid for millions at risk of starvation, say they now fear a siege on the city, where only one major exit route is still open to traffic, on the northern edge of the city.
A military official in Hodeida confirmed seven civilians had died, without giving further details.
A 15-year-old boy died last week of shrapnel wounds in Hodeida, Save the Children said.
Residents reported Tuesday that the fighting had slowed overnight, and rebel media - which regularly claims attacks on loyalists - did not report any new fighting.
The Huthi rebels and Saudi-led coalition bring very different strengths to a complex conflict.
The workers said the Huthis had planted explosives near two of the port's gates, one that leads to Jizan Road, a main street in the city's north, and the other near the Alsanabel flour mill company. Critics of the Saudi campaign have always been questioning U.S. involvement in the war, in which more than 10,000 people have been killed, over 2 million displaced and has spread starvation in Yemen since it began in 2015.
Hodeida port is under a near-total blockade by Saudi Arabia and its allies, who accuse Iran of smuggling arms to the Huthis. Hunt said diplomacy remains the only way to end the conflict, which he called the world's worse humanitarian crisis.
There have been global outrage over the murder of US-based Saudi journalist Jamal Khashoggi, and after Democratic and Republican politicians had threatened to take action in Congress next week over the refuelling operations.
The Yemeni government, supported by Saudi Arabia, and the Houthi rebels, affiliated with Saudi arch enemy Iran, have been fighting for almost four years.
The coalition has been battling Iran-aligned Houthi rebels since March 2015, in a war that has killed tens of thousands of people. But rights groups believe the toll may be five times as high.