It’s way too early for New York Senator Hillary Clinton and her team to celebrate, but the former First Lady has gained back some lost ground in the state of New Hampshire.

The latest Rasmussen Reports telephone poll in the state finds Clinton with support from 31% of Likely Primary Voters while Barack Obama has earned the vote from 28%. Those figures are the reverse of last week’s total and reflect a net six point gain for Clinton. In late November, Clinton had a seven-point lead in New Hampshire.

The latest poll found little change in support for other candidates--John Edwards now attracts 18% support, Bill Richardson 8%, Dennis Kucinich 3% and Joe Biden 2%. Mike Gravel and Chris Dodd split a point between them and 8% are undecided.

A separate survey released today shows Clinton also clinging to a very slight lead in Iowa. But, that race is also too close to call. Clinton and Obama are tied in South Carolina but Clinton’s lead nationally has been growing again in the Rasmussen Reports daily Presidential Tracking Poll.

In the current survey, Obama has a 13-point advantage over Clinton among independent voters likely to vote in the Democratic Primary. That’s little changed from the previous survey. For Obama to be competitive in New Hampshire, he will need a strong turnout from Independents. In a sense, he will now be competing with John McCain for some of these voters since Independents have a choice between voting in the Republican or the Democrat Primary. McCain, recently endorsed by Joe Lieberman, has pulled to within a few points of Mitt Romney in New Hampshire’s Republican Presidential Primary.

Clinton leads based upon strength among older voters, women, and Democrats.

Sixty-eight percent (68%) of Clinton’s supporters are “certain” they will vote for her. That figure is matched by 68% of Obama’s supporters while 60% of those who plan to vote for Edwards are that certain.

Edwards is the second choice for 28% of New Hampshire’s Likely Democratic Primary Voters. Obama is the second choice for 20%, Richardson for 15% and Clinton for 13%.

Nine percent (9%) of Likely Voters say there’s a good chance they could change their mind prior to voting on January 8. Among these voters, Edwards is the second choice for 31%, Obama for 24%, and Clinton for 18%.

Eighty percent (80%) of Likely Democratic Primary Voters in New Hampshire say that Health Care is a Very Important voting issue. Seventy-five percent (75%) say the same about the economy, 71% attach the same importance to Government Ethics and Corruptions, and 70% say Iraq is a Very Important voting issue.

Eighty-one (81%) of New Hampshire’s Likely Democratic Primary Voters have a favorable opinion of both Obama and Edwards. Clinton is viewed favorably by 73%. Those figures include 47% with a Very Favorable opinion of Obama, 41% with a Very Favorable opinion of Edwards, and 39% with a Very Favorable opinion of Clinton. For Clinton, that latter figure reflects a six-point gain over the past week.

Richardson is viewed favorably by 65%, Biden by 57%, and Kucinich by 42%.