While Mr. Jones just arrived from Alabama, most of the others have been venting their frustrations over the polarized nature and inertia of the Senate. They share a desire to show that Congress can work if given the opportunity for free-flowing debate and contend that party divisions can be bridged.

“Maybe it’s an alternative to some of the constant tribal warfare that goes on around here,” said Senator Joe Donnelly of Indiana, another of the red-state Democrats in the group who will be fighting for re-election this year.

Their independent push to end the shutdown sprang out of a view that the two respective Senate leaders — Mitch McConnell, the Kentucky Republican, and Chuck Schumer, the New York Democrat — were locked in a partisan staredown that was leading nowhere.

Leadership was just “ jousting back and forth,” said Senator Joe Manchin III, the West Virginia Democrat who, along with Ms. Collins, played a primary role in assembling the group.

Members of the coalition say the present Senate frequently operates too rigidly along party lines.

“My critique of this place is that everything gets run by the caucuses,” Mr. Kaine said, “And of course I want the caucuses to be strong and I strongly support my caucus leader. But there’s got to be some noncaucus space where people can work together, not just with folks in the same party.”

In recent years under both Democratic and Republican rule, the Senate has been a constrained institution, with highly centralized decision making by leaderships that miserly limits amendments to prevent their members from having to take politically tough votes. With just a few exceptions, the major legislative packages that have reached the floor were the product of the majority and any fights were over Senate process rather than substantive policy.