The Financial Conduct Authority (FCA) also wants it to be easier for people to find the best mortgage, as about 30% fail to find the cheapest deal.
However, it said competition in the market worked well for many people.
The FCA's comments came in an interim report into the mortgage market.
Christopher Woolard, director of strategy and competition at the FCA, said: "For many, the market is working well, with high levels of consumer engagement.
"However, we believe that things could work better with more innovative tools to help consumers.
"There are also a number of long-standing borrowers that have kept up-to-date with their mortgage repayments but are unable to get a new mortgage deal; we want to explore ways that we, and the industry, can help them."
Some mortgage holders found themselves trapped in their current deal when stricter affordability checks on mortgage applications were brought in during 2014.
These "mortgage prisoners" were unable to move to a better deal when their existing mortgages switched back to the more expensive standard variable rate, even if they could meet the payments.
The FCA said it had identified about 150,000 such customers. Of these, about 30,000 were with authorised mortgage lenders, while about 120,000 had mortgages held by non-regulated firms - which include some previous Northern Rock and Bradford & Bingley customers.
The regulator said it intended to explore options to help these customers, "for example, an industry-wide agreement to approve applications for a new mortgage deal from existing customers whose most recent mortgage was taken out before the financial crisis and who are up-to-date with payments".
The regulator said there were high levels of choice and consumer awareness in the mortgage market, with three-quarters of borrowers switching to a new deal within six months of moving on to a standard variable rate.
But it said there was no easy way for customers to be confident of which mortgage deal they might qualify for, and this was "a significant impediment" to shopping around.
The FCA also said a "significant minority" - about 30% - of customers failed to find the cheapest mortgage.
It wants to make it easier for borrowers, early on in the process, to see what mortgage products they can qualify for, and to assess and compare these products.
The FCA is consulting on its interim findings and proposed remedies, and intends to publish its final report around the end of the year.
Jackie Bennett, director of mortgages at UK Finance, said: "Today's interim report highlights that, in the main, the mortgage market is working effectively for the vast majority of borrowers.
"We note the FCA's points regarding perceived areas of weaknesses within the market, particularly around customers who currently may be unable to switch products.
"We will be working through the FCA's recommendations and continuing to engage closely with the regulator over the coming weeks as we respond to the consultation."