The Israeli military issued new evacuation warnings for more than 20 towns and villages in southern Lebanon on Monday, and Hezbollah unleashed a large rocket barrage into northern Israel, the latest indications that the conflict showed few signs of abating.

The widespread warnings across the country’s south, the first in nearly a month, called on civilians to immediately evacuate their homes and move north above the Awali River, farther from the Israeli border. The river effectively demarcates southern Lebanon, which Israel invaded last month in a bid to destroy Hezbollah’s infrastructure and stop it from firing rockets and missiles into Israel.

As that ground offensive continued on Monday, Israel’s new foreign minister, Gideon Saar, signaled that renewed U.S.-brokered diplomatic efforts were now underway to stem the conflict.

“There is progress,” said Mr. Saar, speaking at a news conference. “The main challenge eventually will be to enforce what will be agreed.”

Repeated rounds of shuttle diplomacy over the past year, led by the Biden administration, have so far failed to contain the conflict between Israel and Hezbollah, which began last October when Hezbollah started its cross-border assaults in support of Hamas in Gaza. The violence has significantly escalated in recent weeks. Officials say 3,200 people have been killed and more than a fifth of the population displaced.