Since Lazio replaced New York City Mayor Rudy Giuliani as Mrs. Clinton's GOP opponent on May 20, the poll finds more Empire State voters saying they're undecided - and finds many who've already chosen a candidate saying they'll consider switching.
In the poll, Clinton leads Lazio, 44 percent to 39 percent. But the first lady's advantage over her Republican opponent - be it Giuliani before he quit or Lazio now - has narrowed from eight points to five points, while the percentage of undecideds has risen from 10 to 17 percent. And, the percentage of committed voters who said they won't change their minds has fallen to 61-percent, down from 67-percent in April.
What's good news for Mrs. Clinton is her favorable rating has risen since April - and so has the percentage of people who think she can represent New York effectively even though she recently settled in the state. But Clinton's favorable rating is only 45 percent and her unfavorable rating has held steady at 31 percent for a few months.
More undecideds "means a lot can happen between now and November, and also may be one of the reasons voters think the race is interesting," said CBS News Director of Surveys Kathy Frankovic. In fact, 70 percent of New York voters find the Senate race "interesting" where only a third of voters surveyed nationwide last month found the presidential race interesting.
In upstate New York, which means everywhere north and west of New York City and its suburbs, Lazio has taken a quick 11-pointlead, where Giuliani - who personified the city to many upstate voters - ran close to Clinton and had rarely campaigned. In the Big Apple suburbs, voters still favor the Republican candidate by a wide margin, but now there are more undecideds, too.
The fight for undecided voters may begin upstate, with Catholics.
"Lazio has some built-in advantages which will help him win some of the voters that might have gone to Mrs. Clinton," said Mitchell Moss of New York University. "We haven't elected a white Protestant statewide since Nelson Rockefeller! More than forty percent of the voters are white Catholic." Lazio is a Catholic with an Italian immigrant grandmother; Clinton is Methodist.
And according to ex-Giuliani strategist and pollster Rick Wilson, strong upstate turnout will be important to Lazio because upstate New York is home to the state's Republican base and to ethnic Catholic Democrats who cross party lines. "The big question is," said Wilson, "how motivated can Lazio make them and how much turnout can he achieve?"
Even though Clinton enjoys a 67 to 22 percent lead in the Big Apple, the Democrat cannot take the city for granted. "She needs a huge turnout of Democratic voter to give her a chance to win," said Moss. For Clinton to win, he said, "You have to carry New York City by a large margin" - 300,000 to 400,000 votes - "because you're going to lose the suburbs and you're going to lose upstate."
Despite Lazio's eight-year career in Congress, Clinton comes off as more ready for political prime time in the new poll.
Asked whether the candidates spend more time "explaining positions" or "attacking opponent," 74 percent said Clinton spends more time explaining her positions. In Lazio's case, voters split about one-third apiece for explaining his positions, attacking Hillary and "don't know."
At campaign stops and on the Sunday morning talk shows, Lazio spent the first couple weeks of his campaign reminding voters that he's a lifelong suburban New Yorker who married a local girl - and contrasted his own "mainstream" record with Clinton's "liberal" background.
The effort to paint Clinton as too liberal is misspent energy, said Wilson. "What Lazio should attack is her cowardice: her refusal to appear before the press her fear of being in any environment that is unscripted and uncontrolled and her absolute abject terror of having early debates. If I were Rick Lazio, I would follow her around," challenging her to debate.
Moss said debates will be important to the expanding circle of undecided voters. "We've never seen the two of them in a face-to-face encounter," he said. "I think most voters who are undecided are going to wait to decide until we see how they measure up against each other."
But what happens there may be as much about style as substance, added Moss.
"People are not going to be voting on Lazio's voting record. They're going to be voting on his looks, his personality, his temperament. This is not going to be about issues. It's going to be about personality."