The Financial Conduct Authority (FCA) has found no evidence of an industry-wide or systemic failure to provide customers with sufficient information about enhanced annuities through non-advised sales.
That is among findings of its thematic review of non-advised annuity sales practices.
It instigated the review to establish whether firms gave customers sufficient information about enhanced annuities and encouraged them to shop around to potentially secure a higher income.
The review looked at more than 1,200 non-advised sales of annuities made between May 2008 and April 2015 at seven firms which between them account for some two-thirds of this market.
Many firms provided clear and comprehensive information "with written communication tending to meet the standards required", it found.
But the FCA had concerns about cases where significant communications took place over the phone "which was likely to have caused some customers to purchase a standard annuity when they may have been eligible for an enhanced product".
These failings worried the FCA enough for it to ask those involved to review all non-advised sales from July 2008, and where appropriate provide redress.
Firms involved are also being investigated by the FCA's enforcement division to determine whether further action is necessary.
FCA director of supervision - investment, wholesale and specialist - Megan Butler said: "While we have found particularly poor behaviour at a small number of firms, there is no evidence that firms have systemically failed to provide customers with the information required by our rules.
"Firms, particularly those outside our sample, should look at the report we have published today and consider whether they can make improvements."
The FCA highlighted concerns it had found to the industry to take action, these included:
* call handlers sometimes being heavily reliant on scripts, which meant that they were often unable to respond to clients' needs or clarify misunderstandings;
* customers were not always made aware they could obtain a higher income by shopping around;
* clear messages about enhanced annuities were sometimes undermined by subsequent comments which included call handlers under-playing the level of increase which a consumer may obtain by shopping around;
* firms that did not sell enhanced annuities sometimes failed to mention these products to customers.
The FCA will oversee a further review in which a small number of the largest firms not in its original sample will review their non-advised annuity sales.
Lansdown Hargreaves head of retirement policy Tom McPhail said: "The FCA's review has given the non-advised annuity sector a reasonably clean bill of health however they also found some failings which for a minority of customers may eventually lead to their getting some compensation.
"They also found that even where insurance companies follow the letter of the regulations, they still don't always communicate effectively with their customers.
"If you are buying an annuity you absolutely must shop around. In a small number of cases your existing pension provider may actually be able to offer you the best deal on the market but the chances are that they won't."