Sen. Tom UdallThomas (Tom) Stewart UdallOvernight Energy: Trump temporarily lifts shipping restrictions for Puerto Rico relief New Mexico Gov: GOP health care bill 'still needs some work' Dems ask FEC to create new rules in response to Russian Facebook ads MORE (D-N.M.) on Tuesday became the latest Senate Democrat to back a legislative fix to ObamaCare that would let people keep their health insurance plan if they like it.

Udall signed up as a co-sponsor to the Keeping the Affordable Care Act Promise Act, S. 1642. That bill would allow insurance companies to keep offering insurance plans that would otherwise have to be canceled because they fail to meet new standards under ObamaCare.

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Udall joined the bill without issuing a formal statement, but Senate Democrats are becoming increasingly vocal about the need to change the law. When Sen. Mary LandrieuMary LandrieuYou want to recall John McCain? Good luck, it will be impossible CNN producer on new O'Keefe video: Voters are 'stupid,' Trump is 'crazy' CNN's Van Jones: O'Keefe Russia 'nothingburger' video 'a hoax' MORE (D-La.) introduced the bill earlier this month, she said Democrats need to pass legislation to make good on their promise that Americans could keep their health insurance if they like it.

"We said to people that if they have insurance they like, they can keep it," Landrieu said. "We didn't say that if they have insurance they like that doesn't meet the standards or that meets the minimum standards, they can keep it."

Last week, President Obama announced an administrative fix to the bill that allows state insurance commissioners to continue offering those plans that are due for cancellation for another year. But that led to criticism from some insurance companies that said it would be too difficult to reverse course at the last minute and continue these plans.

In addition, some state commissioners have said they would not use this flexibility offered by Obama. That would appear to mean that people in those states would not be allowed to keep their preferred insurance, contrary to Obama's "keep your health plan" promise.

The addition of Udall to the bill only puts more pressure on Senate Democrats to allow a vote on the measure. But so far, Senate Majority Leader Harry ReidHarry ReidChris Murphy’s profile rises with gun tragedies Republicans are headed for a disappointing end to their year in power Obama's HHS secretary could testify in Menendez trial MORE (D-Nev.) has not said anything about a vote.

Last week, the House passed a similar measure and had the help of 39 Democrats.

Just yesterday, Sen. Joe ManchinJoseph (Joe) ManchinOvernight Energy: EPA aims to work more closely with industry Overnight Finance: Lawmakers grill Equifax chief over hack | Wells Fargo CEO defends bank's progress | Trump jokes Puerto Rico threw budget 'out of whack' | Mortgage tax fight tests industry clout Lawmakers try again on miners’ pension bill MORE (D-W.Va.), kept up the pressure for a legislative fix by saying the Obama administration "messed up royally" with the rollout.

Most of the supporters of the legislative fix are up for reelection in 2014: Landrieu herself, plus Sens. Kay HaganKay HaganLinking repatriation to job creation Former Sen. Kay Hagan in ICU after being rushed to hospital GOP senator floats retiring over gridlock MORE (D-N.C.), Jeff MerkleyJeffrey (Jeff) Alan MerkleyOvernight Energy: Dems take on Trump's chemical safety pick Dem senator slams Trump for dedicating golf trophy to hurricane victims Dem senator compares Trump to Marie Antoinette MORE (D-Ore.), Mark PryorMark PryorMedicaid rollback looms for GOP senators in 2020 Cotton pitches anti-Democrat message to SC delegation Ex-Sen. Kay Hagan joins lobby firm MORE (D-Ark.), and Udall.

Other Senate co-sponsors are Manchin and Sens. Dianne FeinsteinDianne Emiel FeinsteinGun proposal picks up GOP support Gingrich: Banning rapid fire gun modification is ‘common sense’ House bill set to reignite debate on warrantless surveillance MORE (D-Calif.) and Joe DonnellyJoseph (Joe) Simon DonnellyKoch-backed group targets red-state Dems on tax reform Dems plan to make gun control an issue in Nevada Agricultural trade demands investment in MAP and FMD MORE (D-Ind.) — these three senators don't face reelection until 2018.