Ulster Bank is opening 50 branches today from 10am to 3pm, while 10 branches will be open tomorrow from 10am to 1pm.
Despite the progress that the bank is reporting, the widespread problems are expected to continue into next week.
According to the most recent estimate, it will be at least July 16th before the problems are resolved.
The bank admitted last night that it has been double charging some of its customers for transactions in recent days as it works to resolve a technical failure that has left 600,000 customers without access to funds for close to three weeks.
The bank did not say how many accounts had been affected by its latest error, but it apologised to customers and said they would not be left out of pocket as a result of the double charging.
"As we work to restore normal service, some customers may have noticed duplicate transactions on their accounts," a spokeswoman told The Irish Times.
"We apologise for this, and can assure customers that this is being rectified, and that all duplicate postings will be reversed."