groups to vow they would not allow the attacks to go unanswered.
The latest escalation began on Tuesday evening, shortly after the end of a high-profile visit to Gaza by the emir of Qatar, when militants fired six rockets at southern Israel.
Israel immediately carried out twin air strikes over northern Gaza, killing two Hamas militants and injured another seven people, medics said, sparking more rocket fire in response.
Another Israeli raid early on Wednesday killed a third militant and injured another near the southern city of Rafah, medics said.
During the morning, Gaza's Hamas-run health ministry also said another militant who had been seriously wounded in the late-night air strikes had died of his injuries.
Updated figures from the Israeli military issued around midday showed 68 rockets had been fired at Israel since midnight (2200 GMT), leaving six people injured, two seriously and four lightly.
Police confirmed that two Thai nationals working in the fields near the southern Gaza border were "seriously to critically wounded" while a third sustained light injuries.
The other injured were all Israelis.
The military also said its Iron Dome anti-missile system had brought down seven rockets fired towards the southern coastal city of Ashkelon.
Hamas militants and the Popular Resistance Committees (PRC) claimed responsibility for the rocket fire, but did not specify how many projectiles they had fired.
Israel "must strengthen its deterrence against Hamas by attacking the heads of this terrorist organisation or by destabilising its rule in the Gaza Strip," deputy foreign minister Danny Ayalon told public radio.
In Gaza City, Hamas also expressed anger over the spike in violence, with spokesman Fawzi Barhum saying: "The occupation is responsible for every drop of blood in this aggressive escalation."
Gaza's Hamas-run health authority said the early-morning strike on Rafah killed 23-year-old Mohammed al-Sheikh, with the PRC confirming he was one of their militants.
Later in the morning, medics said Yussef Abu Jalhum, 28, had died from injuries sustained on Tuesday night, with Hamas's armed wing, the Ezzedine al-Qassam Brigades, saying he was one of their members.
His death raised to four the number of militants killed within a 12-hour period, three of them Hamas militants. Another 10 people were injured in the strikes.
The Israeli military said that Tuesday night's air strikes targeted rocket-launching squads who were about to fire on southern Israel.
It also said the morning attack on Rafah targeted a squad "moments after it fired rockets toward southern Israel."
Throughout the morning, Israeli aircraft again raided northern Gaza, targeting a "terror tunnel" and other sites, as ground troops fired at "terror activity sites" -- in some cases with tank shells, a statement said.
Palestinian security sources confirmed two air raids, but said there were no fresh casualties.
Tensions have been high in and around the Gaza border for the past few weeks, but peaked early on Tuesday when an Israeli soldier was severely injured by a roadside bomb by the fence in central Gaza.
That attack was claimed by the armed wing of the leftist Popular Front for the Liberation of Palestine.
During the evening, militants fired six rockets over the border, prompting the first of Israel's air strikes.
"The way to fight terror is to fight terror, and that we shall do with great force," Israeli Prime Minister Benjamin Netanyahu warned on Tuesday.