The US military has started reinforcing its positions around oil fields in eastern Syria, saying the new deployments are part of its continuing counter-terrorist mission after the killing of the Islamic State leader, Abu Bakr al-Baghdadi.

General Mark Milley, the chairman of the joint chiefs of staff, said on Monday that US forces would remain in Tanf along the Iraqi border, and more were being sent to the oil fields around Deir Ezzor in eastern Syria.

Video footage from the region showed US military convoys re-entering Syria, days after Donald Trump had ordered them out in advance of a Turkish invasion. The change of mind reportedly came after Pentagon officials persuaded the president that it was essential to protect east Syrian oil resources.

“Keep the oil, I’ve always said that. We’ve secured the oil,” Trump told a police chiefs’ conference on Monday. The president has said that his aim would be to secure a US share of Syrian oil revenues, which is potentially a war crime.

The defense secretary, Mark Esper, claimed that the objective of the deployment was to guard the oil fields to prevent the revenues benefiting Isis but also to keep them out of Syrian regime or Russian control, so that the benefits went to the Kurdish-led Syrian Democratic Forces (SDF).

“We want to make sure that SDF does have access to those resources in order to guard the prisons, in order to arm their own troops in order to assist us with the defeat Isis mission,” Esper said.

Trump has defended his decision to abandon the SDF, saying it was not America’s job to police a longstanding conflict between Turks and Kurds, but Esper said that the US troops remaining in Syria those who remain would be “staying in close contact with the Syrian Democratic Forces, who have fought alongside us”.

He would not say whether the deployment meant that there were more or less US soldiers in Syria than before Trump ordered the withdrawal last month, paving the way for Turkey’s invasion into north-eastern Syria. But the defense secretary insisted that: “At the end of the day we will be sending troops home.”

Esper and Milley gave few new details about the Saturday raid on a compound near the Syrian-Turkish border which led to Baghdadi’s death when he detonated a suicide vest in a tunnel.

“Baghdadi’s remains were then transported to a secure facility to confirm his identity with forensic DNA testing, and the disposal of his remains has been done, and is complete, and was handled appropriately,” Milley said. He would not confirm a Fox News report that the remains had been buried at sea, like those of Osama bin Laden in 2011.

He said that two men had been captured alive at the compound and were in US custody in an unspecified location.

Milley said he could not confirm Trump’s claim that Baghdadi had been “whimpering and crying” as he fled US forces in a tunnel at the compound.

“The president had planned a talk down to the unit and unit members,” the general said. “I don’t know what the source of that was, but I assume it was talking directly to unit.”

The husband of Baghdadi’s wife’s sister has emerged as one of the key figures in the hunt for the Isisi leader. According to Iraqi intelligence officials, Muhammad Ali Sajid al-Zobaie led them to a desert tunnel on the Iraqi Syrian border, where items linked to the Isis leader were found. The Iraqis say they found a weapon, medical bags, religious books and a small handbag containing maps and hand-drawn locations inside the cache.

From that starting point, the smuggler who proved central to the hunt was located, as was one of Baghdadi’s wives and a nephew. All three were co-opted, and they led Iraqis and Syrian Kurds who had then been enlisted by the CIA to northern Idlib province, near an area called Jebel al-Druze in the beginning of September.

Baghdadi is thought to have not long been in the province and was regularly moving between a series of simple houses.

Separately, Kurdish intelligence efforts pinpointed the location of Isis spokesman, Abu Hasan al-Muhajir near the town of Jarabulus on the Turkish border. Al-Muhajir was killed by a US airstrike on Sunday afternoon, only hours after Baghdadi’s demise.

Mazloum Kobani, the commander of the SDF, has claimed that its cooperation was vital to tracking down Baghdadi. He told NBC News that the SDF had a source inside the Isis leader’s inner circle, who he described as a security adviser and one of the few people allowed to visit Baghdadi during his life on the run. Mazloum said this source was able to provide the location and description of the compound where he was killed, including the layout of tunnels.

Mazloum claimed that the SDF had provided a DNA sample for Baghdadi, taken from his underwear and blood. He also said a SDF member took part in the US raid and in a joint raid on Jarabulus in northern Syria, saying that there would be more such raids on Isis militants in “the next few days”.

While not commenting on the SDF’s wider role, Milley said that “the aircraft coming, the aircraft overhead and the soldiers conducting assault was a US only operation”.

• This article was amended on 31 October 2019 to omit a reference to the US energy corporation ConocoPhillips, which no longer operates the gas plant – still sometimes referred to as Conoco - near Deir Ezzor.