SEVERAL MEMBERS of the MBTA’s oversight board called for a major overhaul of the transit system’s fare structure, including a much more serious look at the implementation of charging customers based on how much income they earn, often referred to as means-tested fares.

At a meeting of the T’s Fiscal and Management Control Board on Monday, the members pressed transit authority officials to rethink how the T charges its customers and to resuscitate the idea of means-tested fares. The board appeared to embrace the idea of means-tested fares a year ago, but the policy gained little traction at the T, particularly after Gov. Charlie Baker in January vetoed a provision in transportation bond legislation that would have authorized the T and regional transit authorities to charge customers based on their income levels.

In his veto message, Baker said the idea needs a lot more study. “No means-tested fares can be implemented until the MBTA and RTAs have a financially sustainable plan in place to replace the lost revenue,” he wrote.

Chrystal Kornegay, a member of the control board, said at Monday’s meeting that the T needs to more fully explore means-tested fares, partly as a way to blunt the growing push to do away with fares entirely.

“There’s a lot of movement around free bus and things like that,” Kornegay said, apparently referring to proposals by Boston officials and Sen. Joseph Boncore of Winthrop, the Senate chair of the Legislature’s Transportation Committee. Kornegay said she is not a fan of free bus, since it does away with fares even for those passengers with the means to pay them.  “It makes more sense to do a means-tested fare,” she said.

Another reason the T is reluctant to do away with fares is the transit authority is developing and preparing to install a billion-dollar cashless fare collection system that would give the agency the ability to use one system to collect fares across all transportation modes and provide a lot more flexibility in setting prices.

Monica Tibbits-Nutt, another control board member, backed more study of means-tested fares, but she also directed T staff to explore the idea of doing away with fares entirely for a week in the fall.

“It is a concrete way to show our appreciation for our riders,” she said.

The discussion about free and means-tested fares surfaced during a report to the board on a number of commuter rail fare experiments. For example, the T offers monthly commuter rail passes and daily tickets, but during the pandemic it tried out a flex pass offering passengers five rides at a discount of 10 percent off the regular price. The flex pass generated 6 percent of the T’s commuter rail revenue, but it is being discontinued 90 days after the end of the state’s COVID-19 emergency.

The T also experimented with lower fares on commuter rail service from Brockton and Lynn into Boston to divert passengers away from crowded buses. The experiments attracted 99 new commuter rail passengers in Brockton and only eight in Lynn. The experiments are being discontinued May 31 in Brockton and June 30 in Lynn.

Joseph Aiello, the chair of the Fiscal and Management Control Board, said COVID and the embrace of remote work have scrambled who rides the commuter rail system and how often, and the T needs to figure out how to better serve existing riders and attract new ones.

Aiello asked T staff if they were completely rethinking the existing fare structure. “We ought to be doing it if we’re not doing it,” he said. “We’ve got to experiment.”