The country's dominant Houthi militia and its army allies traded artillery fire with Saudia-backed forces in the Dar Saad and al Alam areas, as Arab warplanes bombed the Iran-allied group.
Assisted by the air strikes, local anti-Houthi forces broke months of stalemate in Aden last week by suddenly seizing the airport and then driving the Houthis out of their last redoubt in the west of the city.
Saudi Arabia intervened in Yemen's war on March 26 in an effort to stop Houthi forces taking Aden, the last city nominally controlled by exiled President Abd-Rabbu Mansour Hadi's government.
Riyadh says it wants to restore Hadi to power in the capital Sanaa, which the Houthis seized in September.
Almost four months of air raids and civil war have killed over 3,500 people in Yemen and deepened suffering in the already impoverished nation, especially in Aden which has seen heavy combat.
A Houthi artillery barrage killed 43 people and wounded 173 on Sunday in the Dar Saad district, underscoring the fragile grip of the anti-Houthi forces.
"We're in an operation to complete the extension of our control over the city of Aden and to confront the Houthi presence at its entrances," a leader in a local militia told Reuters by phone.
Hadi's administration and the Arab military alliance are seeking to secure the city and make it a base from which to challenge Houthi control over most of the rest of Yemen.
The president appointed a new governor for the city on Monday and sources among the local fighters said a technical team from the United Arab Emirates had arrived to repair the city's battle-damaged international airport.
Officials in the anti-Houthi forces say their offensive had been planned for weeks and benefited from training and arms deliveries from Saudi Arabia and the UAE.
The Emirates' official news agency WAM said on Tuesday that an officer from the country had been "martyred" in the service of the Arab campaign, but did not specify where.
The death of another UAE soldier was announced last week, and anti-Houthi militia sources said he died in Aden while advising the local fighters.
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