Despite the support of a Senate majority, Republican filibusters killed the first three attempts at passing jobs bills over the last several weeks, including popular proposals for infrastructure investments and jobs for teachers, police officers, and fire fighters.

The good news is, the Senate managed to pass a component of the American Jobs Act yesterday. The bad news is, it’s an extremely small component; it won’t have a significant impact; and the only reason it passed is because Democrats managed to pay for it while leaving millionaires and billionaires alone.

On the eve of Veterans Day, the Senate approved new measures to help unemployed former service members, advancing a modest piece of President Obama’s $447 billion jobs package with rare bipartisan support.

The bill, approved Thursday by a vote of 95 to 0, would extend tax credits to businesses that hire unemployed veterans. It would also provide new dollars for retraining older unemployed veterans for high-demand fields and includes programs designed to make it easier to get civilian certifications for military training. […]

The bill approved Thursday included another small piece of Obama’s proposal: It would repeal a tax provision slated to go into effect in 2013 that would have withheld 3 percent of payments from government agencies to their vendors.

The measures are paid for, not with surtaxes, but with an extension of an existing VA fee that had been set to expire.

I don’t want to dismiss the tax credit that encourages the hiring of veterans as insignificant. The unemployment rate among vets is roughly three points higher than the civilian population, and any measure intended to help is welcome.

But as a practical matter, even the most optimistic scenario suggests the impact of yesterday’s measures will be very modest, and the end of the 3% withholding rule — which Republicans hate, despite having created it in the first place — is expected to make a “miniscule” difference.

It’s why the self-congratulatory rhetoric on Capitol Hill yesterday was a little hard to take. Republican senators seemed eager to boast about how cooperative they were being, using the votes as proof that they’re willing to occasionally play a constructive role.

But the larger context makes all the difference. The only jobs bills that Republicans will even consider are those that (a) don’t create a lot of jobs; and (b) shield millionaires and billionaires from even the most modest sacrifices.

The House is expected to approve the Senate’s measures, which the White House also supports.

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Follow Steve on Twitter @stevebenen. Steve Benen is a producer at MSNBC's The Rachel Maddow Show. He was the principal contributor to the Washington Monthly's Political Animal blog from August 2008 until January 2012.