The US and Iran have inched closer to all-out war as last month's interim deal meant to permanently end the fighting has crumbled and shipping traffic in the Strait of Hormuz has stalled.
Both sides have targeted civilian infrastructure relied on by millions of people.
The US military said the service member was killed in Iraq on Saturday during the "controlled detonation" of a downed Iranian drone.
Earlier, the military said its strikes targeted Iran's paramilitary Revolutionary Guard to retaliate for Friday's killing of US troops in Jordan.
One service member was missing after that attack, and the military's new statement said "unidentified remains" were found on Sunday and were being examined. Since the war began, 17 US service members have been killed.
The renewed exchanges of fire, now in their second week, have seen the US target bridges and electrical facilities in Iran.
The US military's Central Command announced a new round of strikes early on Monday, now into their ninth straight night.
"The strikes will continue degrading Iranian military capabilities used to attack commercial vessels and civilian mariners transiting the Strait of Hormuz," Central Command said.
Tehran has retaliated by hitting US-allied countries throughout the Middle East.
Bahrain, Jordan and Kuwait again activated air defences for incoming Iranian drones and missiles. Israel warned that missiles launched toward neighbouring Jordan could cause fire to spill over into Israeli territory for the first time in weeks.
Footage released on Sunday by the US military appeared to show strikes carried out by fighter jets and by Tomahawk cruise missiles launched from the sea. One target site appeared to be in a valley of a mountainous region. The Guard often has missile bases and other military equipment tucked into mountain ranges.
Iran's atomic energy agency said US strikes targeted the construction site of a planned nuclear power plant in the southwest, state television reported.
Satellite images of the Darkhovin nuclear plant site showed earth clearing but little construction as of July 9. Iran did not previously announce it as being targeted.
The International Atomic Energy Agency said the site, in the "very early stages of construction", contained no nuclear material when the UN watchdog last visited.
Jordan's military said it shot down several Iranian missiles. The country hosts major US bases and relies on US air-defence systems.
Israel's military said Iranian missiles launched toward the Jordanian port city of Aqaba just across the border could spill over, warning Israelis to expect the first air-raid sirens in weeks.
During the latest fighting, Iran has focused attacks on US-allied Arab states rather than Israel.
"We are prepared to immediately resume combat," Israel's military chief of staff, Lieutenant General Eyal Zamir, said.
Kuwait said one of its power and water desalination plants was attacked for the second time in two days, causing fires.
Its Ministry of Electricity, Water and Renewable Energy said the power grid remains stable. In Kuwait, about 90 per cent of drinking water comes from desalination.
The secretary-general of the six-nation Gulf Cooperation Council, Jasem Mohamed al-Budaiwi, has accused Iran of war crimes for strikes on infrastructure and civilian facilities.
International humanitarian law protects civilian infrastructure such as bridges and power plants from attack, but such sites can lose protections if used for military purposes. In such cases, attacks must be proportionate and minimise civilian harm.
US President Donald Trump has threatened to target Iran's power stations and bridges to try to compel Tehran to loosen its hold on the Strait of Hormuz.
The US in the past week reimposed a naval blockade on Iranian ports to halt its shipments of crude oil. The military on Saturday said it had redirected six ships and disabled one since then.
Iranian Supreme Leader Mojtaba Khamenei, in a statement on Saturday, warned of "unforgettable lessons" if the US kept attacking the Islamic Republic.
