(CNN) Three-in-10 Americans say they would only vote for a candidate for major office who shares their views on abortion, according to a new CNN poll conducted by SSRS . That's higher than at any point in CNN polling on the topic from 1996 on.

Gender is a big factor in whether a person views abortion as a critical issue, even more than partisanship. Women (33%) -- especially independent women (42%) and non-white women (39%) -- are more apt to consider it a critical issue than men (26%). The share who say it is critical for them is about the same across party lines (33% for independents, 29% for Democrats, 28% for Republicans).

Twenty percent overall say they don't see abortion as a major issue, while 45% would consider a candidate's position on abortion, but don't see it as a decisive issue.

After former Vice President Joe Biden said Wednesday that he still supports the Hyde Amendment -- a prohibition on federal funds being used for abortion, except in cases of rape, incest, or when the life of the mother is at stake -- many of his opponents jumped at the opportunity to draw a distinction with the frontrunner and oppose his take.

The poll was conducted before the debate over the Hyde Amendment erupted, and, in its results, Biden's backers were no different from other potential Democratic primary voters on whether the issue is central to their choice for major offices (30% among Biden's supporters, 32% among other potential Democratic voters).

But abortion has been a prominent issue outside of the Democratic contest as well. Georgia, Alabama, Missouri and other states have enacted laws placing sharp limits on abortion designed to test the Supreme Court's Roe v. Wade ruling, which established a woman's right to an abortion.

The poll finds most Americans are opposed to bans on abortion.

In general, Americans' views on when abortion ought to be legal have shifted slightly toward the extreme ends of the scale since the last time CNN polled on the topic in 2016.

A majority felt it should be legal only under certain circumstances in that poll (53%), but now, 48% feel the same way. At the same time, the shares saying abortion ought to be legal in all circumstances (31%) and illegal in all circumstances (18%) have each ticked up a couple of points.

About 4 in 10 whose views on abortion fall at the far ends of the scale say they will only support a candidate who shares their views on abortion (42% among those who say abortion should be illegal in all situations, 37% among those who say it should be legal in all). That drops to 21% among those who think it should only be legal under some circumstances.

Asked more directly about a law in their own state banning all abortions except those necessary to save the life of the mother, nearly 6 in 10 Americans (58%) are opposed, 36% would favor it. Majorities of Republicans (53%) and conservatives (57%) would favor such a law, but opposition among Democrats (75%) and liberals (82%) is stronger.

The CNN Poll was conducted by SSRS May 28 through 31 among a random national sample of 1,006 adults reached on landlines or cellphones by a live interviewer. Results for the full sample have a margin of sampling error of plus or minus 3.8 percentage points. It is larger for subgroups.