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Of particular concern is the margin of loss among Hispanics, a group Mr. Obama won by about 70 percent to 30 percent.

It took only hours after the loss for national GOP leaders to blame Mr. Romney for shifting to the right on immigration — and signal that the party must change.

Mr. Jindal, a prospective 2016 presidential contender, was among the Republicans calling for a more measured approach by the GOP. And even previously hard-line opponents of immigration reform — like conservative talk show host Sean Hannity — said the party needs to get over its immigration stance heavily favoring border security over other measures.

“What you have is agreement that we as a party need to spend a lot of time and effort on the Latino vote,” veteran Republican strategist Charlie Black said.