A spokesman for the Tigrayan forces told AFP they had seized Alamata, the main town in southern Tigray, after launching their latest assault on Monday.
Getachew Reda said fighting was also taking place in western Tigray, an area where the United States has raised concerns about ethnic cleansing.
The rebel claims could not be independently confirmed because communications were largely down in the area, while an Ethiopian military spokesman did not immediately respond to a request for comment.
The Tigray Defence Forces last month swept across large parts of Tigray and seized the regional capital Mekele after eight months of brutal conflict with federal troops.
The fighting — marked by grisly massacres and widespread sexual violence — has killed thousands of people, while the United Nations says hundreds of thousands are on the brink of famine.
Security forces and officials from the neighbouring Amhara region had moved in to both the southern and western areas of Tigray in November in support of the Ethiopian army, after Tigrayan forces cleared out during the early phase of the war.
“We promised to liberate every square inch of Tigray,” Getachew said.
“Yesterday (Monday) we launched an offensive in (the southern region of) Raya and were able to absolutely rout federal defence forces and Amhara special forces divisions,” he said.
“We have been able to secure most of southern Tigray including Korem and Alamata (the main town in the area)”.
Getachew said TDF fighters were still “in hot pursuit” of pro-government fighters, adding: “We don’t want to give them a chance to regroup.”
A UN source reported Tuesday the sound of artillery fire near Emba Madre, a town in western Tigray, while to the south, aid workers reported hearing small arms fire near a refugee camp at Mai Aini.
Active fighting was also reported around Mai Tsebri, about 13 kilometres (eight miles) from Emba Madre.
A local government office in the Amhara region to the south of Tigray had called on Monday for a “mass mobilisation” of people with arms and food to go to Mai Tsebri.
The rebel offensive was launched just two days after election results showed Prime Minister Abiy Ahmed had won by a landslide in a June election that went ahead despite the Tigray conflict that has battered his global reputation.
The TDF had described its seizure of Mekele and most of Tigray as a major victory and initially branded the government’s unilateral ceasefire a “joke.”
Rebel leaders later said they accepted the ceasefire “in principle” but posed strict conditions including the withdrawal from the region of Eritrean and Amhara forces.
Abiy and other officials have countered that federal forces executed a strategic pullback to focus on other threats.
Abiy — who won the Nobel Peace Prize in 2019 for his peace deal with neighbouring Eritrea — sent the army into Tigray last November to oust the region’s once-dominant ruling party, the Tigray People’s Liberation Front (TPLF).
Abiy had accused them of orchestrating attacks on Ethiopian military bases in Tigray, an important economic and industrial region in the Horn of Africa nation.
The war has badly damaged Abiy’s international standing, and Western powers have demanded that the ceasefire be accompanied by unfettered aid access into the desperate region where famine-like conditions have been reported.
EU foreign policy chief Josep Borrell on Monday urged member states to consider imposing sanctions, warning the Tigray crisis “has never been as bad” and that a major humanitarian disaster was looming.
The US, a traditional ally of Ethiopia, reiterated Monday its finding that “acts of ethnic cleansing” had taken place in western Tigray, and called on all armed parties to protect civilians.
Tigrayan officials in March accused forces from neighbouring Amhara of kicking thousands of people off land in western Tigray — a part of the region that ethnic Amharas claim rightfully belongs to them.
Many ethnic Amharas believe the once-dominant TPLF illegally incorporated the fertile territories after it came to power in the early 1990s — and that they should fall under Amhara administration.
The World Food Programme (WFP) said Monday that a convoy of 50 trucks carrying 900 tonnes of relief supplies had arrived in Mekele, but that it was not near enough to help the hunger-stricken region.
“We need double this number of trucks arriving daily, and we need them to take two days to reach Mekele instead of the four-day journey this time if we are to reach the millions of people in need of life-saving assistance,” said WFP emergency coordinator Tommy Thompson.