The latest disclosure campaign runs until March
Savers with offshore accounts who have failed to pay tax owing to an "innocent error" will not face a fine.
HM Revenue and Customs (HMRC) is giving people a second - and last - chance to pay tax on money hidden in offshore bank accounts.
Now it has has confirmed that those misled by financial professionals or who could point to bereavement or serious illness could avoid fines.
But those simply ignorant of tax laws would still face a penalty.
Meanwhile Switzerland has become the latest country to agree to exchange tax information with the UK government.
The Swiss government decided in March that it would abide by standards laid down by the Organisation for Economic Co-operation and Development (OECD).
A spokesman for the HMRC said the signing in London of a protocol between the two countries meant they could exchange tax information about individuals, where a justified request had been made.
"I very much welcome the Swiss Federal Council's agreement on international co-operation in tax matters and their adoption of the OECD standard on administrative assistance," said Stephen Timms, financial secretary to the Treasury.
"The days when hiding money off-shore represented a viable means of evading UK tax are rapidly drawing to a close," he said.
It is not illegal to have offshore accounts, but anyone who receives savings and investment income from abroad needs to declare it on their tax return, as they might have to pay UK income tax as a result.
HMRC is conducting a second disclosure initiative calling on people with an offshore account who have not revealed such income to settle their tax bill.
Those who come forward before 12 March, 2010, will face a fine of 10% of the unpaid tax but HMRC permanent secretary for tax, Dave Hartnett, said that the fine would be ignored in cases of innocent error.
"The new approach to penalties in the HMRC is that for truly genuine mistakes we do not levy penalties," he told BBC Radio 4's Money Box programme.
One example the HMRC gives is when a bank gives an incorrect calculation of the interest received from an account, which led to the amount being paid in tax being too low.
An HMRC spokeswoman said that the tax authority had brought in one penalty system for inaccurate tax documents and returns across the main taxes in April.
Fines would generally not be levied in cases affected by exceptional events beyond people's control, such as the death of a partner or close relative around the time the person should have told HMRC of the taxable activity.
Postal strikes or floods would also offer a "reasonable excuse", but pressure of work, lack of information and ignorance of the basic law were not normally regarded as genuine excuses.
Under existing rules, those with a total amount of unpaid tax and duties of less than £1,000 are not fined.
However, anyone who failed to come forward during the latest disclosure campaign faces a fine, if later caught, of at least 30% of the unpaid tax, but they could also be prosecuted.
More than 300 UK and foreign banks were last month told that they had to hand over details of UK taxpayers who have accounts offshore.
The HMRC also struck a deal to exchange details with the authorities in Liechtenstein, where an estimated 5,000 British investors are thought to have funds in secret accounts in the country.