"Thank you for your patience and for bearing with us", CEO Paul Pester said on Twitter on Wednesday morning.
TSB said on Monday that it was still suffering "intermittent issues" with its services, but didn't go any further than that. It advises customers to: "Please bear with us and keep trying".
But some TSB customers responded on Twitter that their access was still blocked, and that the bank's website as of 0913 GMT said it was "limiting access to internet banking".
MPs on the House of Commons Treasury Select Committee have demanded answers from TSB over an IT meltdown this week, following an attempt to shift customers from Lloyds Bank's banking platform over the weekend.
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Other customers, meanwhile, report erroneous account details, including someone who, overnight, apparently acquired significant arrears in their mortgages - which had also been re-denominated into United States dollars.
Bank TSB is continuing to limit access to its mobile banking app and online service after days of delays that saw customers unable to use their accounts.
The upgrade window was scheduled between Friday at 4pm and Sunday at 6pm.
"It's a very concerning development for any customer of any bank to be anxious about where their money has gone", Irish Finance Minister Paschal Donohoe told reporters earlier on Tuesday, adding that the central bank was reviewing the matter. "Potentially millions of customers could be affected by uncertainty and disruption", she said.
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"Warm words and platitudes will not suffice".
"What surprised me is the fact that TSB allowed the buggy system to run through their 1.9 million customers [without what appears to be] proper testing of the new system", he said.
The Financial Conduct Authority, which has the power to fine banks for such failings, says it is "aware of the issue", while Information Commissioner's Office says it is "making enquiries" about a "potential data breach".
"It's particularly disappointing because the bank has been doing a lot of good things, but you can't just get it right 90% of the time and they are going to have to answer some hard questions".
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Royal Bank of Scotland's Irish division also reported an issue that led to salaries going missing from some accounts on Tuesday.