Senate Republicans cited their minority rights in preventing the “don’t ask, don’t tell” repeal from coming to the floor Tuesday, and they are likely to recycle the message in the run-up to the midterm elections.

“Our goal is to represent the voices of the American people, to let their feelings, their angers, their hopes all be represented here,” Senate Republican Conference Chairman Lamar Alexander (Tenn.) said after the defeat of a procedural vote to take up the defense authorization bill, which included the repeal of the military’s ban on openly gay service members.

“If we’re successful in this election year, we’re going to make sure that in the new Congress, we have that opportunity,” Alexander added.

At issue is the matter of the filibuster. Senate Majority Leader Harry Reid (D-Nev.) has held votes to circumvent an onslaught of GOP amendments to many prominent pieces of legislation this year, including the defense measure. Democrats say the amendments are a form of filibuster designed to indefinitely delay votes on passing bills. Republicans argue that they are being robbed of opportunities to shape legislation on the floor.

Republicans have used this message in the past, most notably in opposing the health care overhaul, and using it now gives them political cover to stall the defense authorization.

Democrats have considered changing filibuster rules at the opening of the next Congress, although those discussions quieted after Members feared it would look like a power grab. But Armed Services Chairman Carl Levin (D-Mich.) told reporters Tuesday that the derailment of the defense measure “surely is a very powerful argument for why we should change the rules of filibuster relative to motions to proceed.”

In floor remarks criticizing Reid’s management of the defense bill, Sen. Susan Collins (R-Maine) said Tuesday, “I cannot vote to proceed to this bill under a situation that is going to shut down the debate and preclude Republican amendments.

“This would be the 116th time in this Congress that the Majority Leader or another member of the majority has filed cloture rather than proceeding to the bill under an agreement that would allow amendments to be debated,” she added.

Collins cited those reasons in voting against cloture to proceed to the defense bill, a measure she supported in the Armed Services Committee, including the DADT provision. Minority Leader Mitch McConnell (R-Ky.) and Sen. John McCain (R-Ariz.), among others, offered similar explanations for opposing the procedural motion. Both are also opposed to the DADT repeal language, arguing that the Pentagon should complete its yearlong review before the Senate casts a vote on the issue.

Reid also wanted the defense measure to include an amendment that would curb the Senate’s use of secret holds. While Sens. Ron Wyden (D-Ore.), Claire McCaskill (D-Mo.) and Chuck Grassley (R-Iowa) have long championed the issue, most Republicans said the amendment should be tabled because it is not relevant to defense matters.

Republican aides said the minority-rights argument could be raised again when the chamber debates extending tax cuts enacted under President George W. Bush and other issues slated for the lame-duck session, including a second attempt at passing the defense authorization measure.

The Rules and Administration Committee, which has held a series of hearings on filibuster reform, will meet Wednesday to hear testimony from Democratic Sens. Tom Harkin (Iowa) and Tom Udall (N.M.), who have proposals for tweaking Senate rules governing the filibuster. Alexander, who sits on the committee and is the GOP point man on rules reform, said Democrats should be wary of any major rule changes, because “the shoe can sometimes be on the other foot.”

“Those who today are wanting to create a freight train running through the Senate, as a freight train runs through the House, might not be so eager to do that if the freight train turns out after the election to be the Tea Party Express,” he said.