Jedburgh became the latest town to receive the provision last week.
Councillor Stewart Bell said by the end of the latest rollout, 94% of homes in the region should have access.
He added that a range of groups were working on ways to deliver superfast services to those not covered by that programme.
There have been concerns in the past that the region was missing out despite major investment in broadband.
Mr Bell said there were technical reasons why some sites would not see their services improve.
"Over very long distances the speeds run down a bit and therefore there are some premises that are very distant from cabinets that won't see a significant improvement in services," he said.
"The target for this superfast scheme which will be fully operational between 2017 and 2018 is to get 94% of premises in the Borders connected.
"The remaining 6% is something that is being considered by Digital Scotland and Community Broadband Scotland in co-operation with local authorities to see if we can come up with customised local solutions."

But the manager of the restaurant, called Halal KFC, said it had nothing to do with the US company.
Police justified the decision saying it was operating under a false licence.
The closure comes amid concerns amongst hardliners about growing Western influence in Iran as relations with a number of countries improve.
"The shutting down of Halal KFC was due to a misunderstanding," Abbas Pazuki, the manager of Halal KFC, told the Tasnim News Agency.
He said police had thought the restaurant was a branch of the American KFC.
"We are part of a brand known as Halal KFC, which comes from Turkey. It belongs to Muslims and its target market is Muslim nations," said Pazuki.
He said the Turkish brand was a "rival of the American KFC".
Earlier reports suggested the authorities closed down the first branch in Tehran of the better known KFC.
Ali Fazeli, head of the Iranian chamber of commerce, confirmed that the Iranian KFC has no connection with KFC in the US, according to ILNA press agency.
"In accordance with orders from the Supreme Leader, we do not give any authorisation to Western brands" in the fast food sector, Fazeli said.
The state media reported the opening of the restaurant as a first sign of creeping US influence, the BBC's Kasra Naji says.
Hardliners believe the recent agreement on curbing Iran's nuclear programme, reached between Iran and Western powers, may become a vehicle for softening Iran's anti-Americanism, opening the floodgates of foreign influences, our correspondent adds.
Several Western countries are seeking closer business ties with Tehran following the agreement.
A Borders councillor has said work is being done to deliver superfast broadband to homes in the region currently missing out on the service.
[NEXT_CONCEPT]
Only a day after it opened, Iran has shut down what officials reportedly thought was a branch of the US fast food giant Kentucky Fried Chicken.