Mrs Clinton said China and India in particular could increase pressure because of their energy investments.
She said the US did not want to call for Mr Assad to stand down without getting the backing of other countries.
Activists say the army killed at least 24 people on Thursday and new protests flared after dawn prayers on Friday.
More than 1,700 people have died and tens of thousands of people have reportedly been arrested since the uprising against the the 41-year rule of President Bashar al-Assad's family began in March.
'Chorus of condemnation' This week, Washington added to its existing sanctions on Syria by including its main commercial bank and mobile phone company, and warned that more could follow.
But in an interview with CBS News, Mrs Clinton said other countries also needed to exercise their influence over the regime.
"What we really need to do to put the pressure on Assad is to sanction the oil and gas industry," she said, citing Europe, China and India as powers with energy investments in Syria and adding that the US also wanted Russia to stop selling arms to the Assad regime.
When asked why Washington has not called yet outright for Mr Assad to stand down, Mrs Clinton said the US was focusing on "building the chorus of international condemnation".
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Amer al-Sadeq, a member of the anti-government Syrian Revolution Co-ordinators' Union, has told the BBC economic sanctions would be a boost to the opposition movement.
"This will help the silent majority take a firmer attitude against the regime and to declare their support more for the peaceful revolution to achieve its goal better," he said.
After talks on Thursday, US President Barack Obama and Turkish Prime Minister Recep Tayyip Erdogan reiterated their "deep concern" about the use of violence against civilians.
"The Syrian people's legitimate demands for a transition to democracy should be met," they said in a joint statement.
Kahn Sheikhun assault The BBC's Jim Muir in Beirut says there were more protests in Syria after dawn prayers on Friday, with activists reporting one person shot dead and others wounded in the Damascus suburb of Saqba.
He says that while Friday's noon prayers are again expected to be the focal point of demonstration, activists are saying that during the current fasting month of Ramadan, each day is like a Friday, with people protesting every night after evening prayers and again after the early-morning prayers.
Syrian troops at dawn on Friday launched an operation in Kahn Sheikhun, in north-western Idlib province, activists said.
Rami Abdel Rahman, of the Syrian Observatory for Human Rights, said one woman was killed as "dozens of tanks, troop carriers and civilian cars" took part in the assault.
Security forces are also reported to be continuing their drive to secure control of the city of Deir al-Zour in the east, moving in on a mosque that has been a centre of defiance. At least three deaths were reported in the city on Thursday.
A day earlier, Syria had allowed international media to film as the army withdrew from the severely attacked city of Hama, a move correspondents said had clearly been aimed at appeasing Damascus's neighbour, Turkey, and other outside powers.
But on Thursday, Syrian security forces continued the crackdown, reportedly killing at least 24 people, mainly in the province of Homs.
Most were reported to have died in the western town of Kassir, 135km (85 miles) north of Damascus, after communications and electricity were cut and tanks and troops swept in.
Activists said many residents tried to flee the town and that a woman and a child were among the dead. They said one person also died in the coastal city of Latakia.
Meanwhile, the BBC's Lina Sinjab in Damascus said many had been wounded and 27 arrested in a crackdown on a sit-in protest by engineers in the southern mainly Druze town of Sweida.
Troops and tanks also moved into Saraqeb, near Syria's north-western border with Turkey, with houses raided and many arrests being made, she adds.
International journalists face severe restrictions reporting in Syria, and it is hard to verify reports.
The recent large-scale operations by the army have drawn international condemnation, with Saudi Arabia, Bahrain and Kuwait recalling their ambassadors from Damascus.
Mr Assad has reiterated promises of political reform, while remaining adamant his government would continue to pursue the "terrorist groups" he has blamed for the unrest.
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