Israel's military pounded targets in south and east Lebanon on Monday and said more strikes were coming, warning Lebanese to stay out of harm's way despite international calls for restraint.
Hezbollah has said it is ready for "all military possibilities".
Rear Admiral Daniel Hagari, the Israeli military spokesman, made a first of its kind appeal to people in Lebanon.
He told them to avoid potential targets linked to the Iran-backed Hezbollah as new strikes were launched Monday that would "go on for the near future".
Hagari said Israel's military "will engage in (more) extensive and precise strikes against terror targets which have been embedded widely throughout Lebanon".
He urged Lebanese civilians "to immediately move out of harm's way for their own safety".
After nearly a year of tit-for-tat cross-border fire between Hezbollah and Israeli forces, exchanges over the weekend reached their most intense since the outbreak of war between Israel and Palestinian Hamas militants in the Gaza Strip last October 7.
Hezbollah says it is acting in "support" of Hamas.
Ahead of the annual General Assembly, United Nations chief Antonio Guterres warned of the risk of Lebanon becoming "another Gaza" and said it was "clear that both sides are not interested in a ceasefire" in the Gaza war.
Lebanon's official National News Agency said Monday "enemy warplanes launched... more than 80 air strikes in half an hour", targeting south Lebanon, at the same time as "intense raids in the Bekaa" Valley area in the east, where it said a shepherd was killed.
AFP correspondents in the south and east reported the sound of heavy strikes.
After intense rocket fire from Lebanon, Israeli Prime Minister Benjamin Netanyahu said Sunday that Israel has dealt "a series of blows on Hezbollah that it could have never imagined".
Hezbollah's deputy chief, Naim Qassem, said the group was in a "new phase" in its battle against Israel.
Both spoke after attacks on northern Israel forced Israelis to seek shelter and caused damage in the area of Haifa, a major city on Israel's north coast.
"No country can tolerate attacks on its citizens," Netanyahu said, as Israel turns its focus on Hezbollah, nearly a year into its Gaza war sparked by Hamas Palestinian militants' attack on Israel.
The Gaza war has drawn in Iran-backed groups across the Middle East, including Hezbollah.
Since the near-daily border exchanges began in October, hundreds of people have been killed in Lebanon, mostly fighters, and dozens in Israel and the annexed Golan Heights.
Tens of thousands of people on both sides have fled their homes.
Israeli Defence Minister Yoav Gallant said Sunday military action "will continue until we reach a point where we may ensure the safe return of Israel's northern communities to their homes".
Israel's ally and leading weapons supplier the United States said military escalation is not in Israel's "best interest".
President Joe Biden said his administration was "going to do everything we can to keep a wider war from breaking out".
The European Union and Britain stressed the need for a ceasefire, and China on Monday urged its nationals to leave Israel, citing the "extremely tense" situation on the northern border.
Despite Israel's sophisticated air defences, Hezbollah rocket fire reached Kiryat Bialik near Haifa, leaving a building in flames, another pockmarked with shrapnel and vehicles incinerated.
"This is not pleasant. This is war," said one resident, Sharon Hacmishvili.
Analysts say Hezbollah has been dealt a serious blow over the past week.
Deadly attacks targeted its communications and decimated the leadership of its elite unit, although its ability to fight has not been crushed, the analysts said.
An Israeli air strike in a densely populated Hezbollah stronghold in southern Beirut Friday killed the head of Hezbollah's elite Radwan Force, Ibrahim Aqil.
Lebanon's health ministry said the strike killed 45 people. The toll included many civilians as well as other Hezbollah commanders.
The Beirut strike came after a series of coordinated communications device blasts on Tuesday and Wednesday across Lebanon killed 39 people and wounded almost 3,000, and which Hezbollah blamed on Israel.
Speaking at Aqil's funeral on Sunday, Qassem said: "We have entered a new phase, namely an open reckoning" with Israel.
"Threats will not stop us... We are ready to face all military possibilities."
Israel's military said more than 150 rockets, missiles and drones were fired at its territory overnight Saturday and early Sunday, most from Lebanon.
It said it attacked Hezbollah targets in southern Lebanon in response and "to prevent a larger-scale attack".
Hezbollah said it had targeted Israeli military production facilities and an air base in the Haifa area as "an initial response" after the communication device blasts.
Netanyahu last week announced an expansion of Israel's war goals to include the return home of northern residents.
International mediators from Qatar, Egypt and the United States have for months tried to secure a ceasefire and hostage release deal in Gaza, which diplomats repeatedly said would help calm regional tensions.
Egyptian Foreign Minister Badr Abdelatty told AFP Sunday the Israel-Hezbollah flare-up "negatively affects" Gaza ceasefire efforts.
"The problem is the lack of political will on the Israeli side," he added.
Netanyahu's critics in Israel have accused him of dragging out the war.
Hamas's October 7 attack on Israel resulted in the deaths of 1,205 people, mostly civilians, according to an AFP tally based on Israeli official figures that include hostages killed in captivity.
Israel's retaliatory military offensive has killed at least 41,431 people in Gaza, most of them civilians, according to figures provided by the Hamas-run territory's health ministry. The UN has acknowledged the figures as reliable.
