Opponents of the Kansas referendum leaned into that “freedom” message, with advertising that cast the effort as nothing short of a government mandate — anathema to voters long mistrustful of too much intervention from Topeka and Washington — and sometimes without using the word “abortion” at all.

Some of the messaging was aimed at moderate, often suburban voters who have toggled between the parties in recent elections. Strategists in both parties agreed that abortion rights could be salient with those voters, particularly women, in the fall. Democrats also pointed to evidence that the issue may also drive up turnout among their base voters.

After the Supreme Court’s decision, Democrats registered to vote at a faster rate than Republicans in Kansas, according a memo from Tom Bonier, the chief executive of TargetSmart, a Democratic data firm. Mr. Bonier said his analysis found roughly 70 percent of Kansans who registered after the court’s decision were women.

“It is malpractice to not continue to center this issue for the remainder of this election season — and beyond,” said Tracy Sefl, a Democratic strategist. “What Democrats should say is that for Americans your bedroom is on the ballot this November.”