The jihadists, who have lost ground in Iraq over the past few months, meanwhile claimed a suicide bombing that killed eight people near a Shiite mosque in eastern Baghdad.
Backed by sustained air strikes from the US-led coalition, elite troops recaptured Ramadi's southwestern neighbourhood of Al-Tameem on Tuesday.
The advance was hailed as a significant step in efforts to retake Ramadi, a key IS hub 100 kilometres (60 miles) west of Baghdad, and fragment the jihadists' self-proclaimed "caliphate".
As the counter-terrorism forces that led the offensive prepared their next move, local forces from Anbar province moved in to hold recaptured territory.
Local police "moved from Habbaniyah to hold the land in Al-Tameem following its liberation," a local councillor in the nearby Khaldiyah area, Ali Dawood, told AFP.
The force consists of around 500 fighters, a military source said.
One of the main tasks for Iraqi forces is to clear the area of bombs planted by IS, a favoured tactic of the jihadists that means they can kill security personnel and civilians long after they have withdrawn from an area.
Cars and trucks laden with tonnes of explosives and driven by suicide attackers were a key weapon in IS's shock capture of Ramadi in mid-May.
The coalition said in a statement that four such car bombs were destroyed in air raids on Tuesday, as well as a plant where they were manufactured.
A total of four coalition strikes in the area also destroyed IS positions and weapons and impeded the movements of jihadist fighters, the statement said.
Iraqi forces spent months cutting off IS supply lines around Ramadi and slowly closing in on the city by taking suburban areas one after the other.
"The liberation of Al-Tameem was very important and... enables other forces to advance toward the centre of the city of Ramadi," said Sabah al-Noman, the spokesman for Iraq's counter-terrorism forces.
He and other military sources said Iraqi forces had also retaken the Anbar Operations Command, a key facility which lies at a fork of the Euphrates River in the city.
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- Concern over civilians -
Speaking in Baghdad, the top US official in the international anti-IS coalition, Brett McGurk, said the presence of civilians in Ramadi will require a cautious approach as the operation moves forward.
"There are still civilians inside Ramadi -- Daesh is not allowing any civilians to leave," he told reporters, using an Arabic acronym for IS.
"They're holding civilians as hostages, and as human shields, and so we want to do this in a very careful and deliberate way," he said.
An Iraqi brigadier general involved in the Ramadi operation said civilians who are able to move are being urged to head to Humayrah, a staging ground controlled by Iraqi forces on the southern edge of Ramadi.
He admitted that it was difficult for those residents still inside the city to leave.
"Daesh (IS) does not allow Ramadi residents to move... those who try to leave the city are executed," he said.
The spokesman for the counter-terrorism forces however said he had no knowledge of any civilians remaining inside the city.
"The information that we have is that there are no civilians inside the city of Ramadi," Noman told AFP.
IS overran large parts of Iraq in June 2014, including major territory in Anbar, which stretches from the borders with Syria, Jordan and Saudi Arabia to the western approach to Baghdad.
Shifting parts of Ramadi had been held by anti-government fighters since the beginning of 2014, but IS did not succeed in completely overrunning it until May of this year.
There was no reaction to the developments in Ramadi from IS on its usual social media channels but the group claimed a suicide attack that killed eight people in the capital Baghdad Wednesday.
The blast near a Shiite mosque in the eastern Obeidi area also wounded at least 19 people, security and medical officials said.