Walker's forceful rhetoric, shortly after the Supreme Court legalized same-sex marriage nationwide, shocked some donors and their representatives -- a message they passed along to campaign staffers, according to multiple people familiar with the conversations. He met disappointment at home, too. His two sons support same-sex marriage.

A day after calling for the amendment -- and pronouncing the Supreme Court's decision a \"grave mistake\" -- Walker offered a different tone in a speech to conservatives in Denver. This time he called for respect on the same-sex marriage ruling. He made no mention of a push for a constitutional amendment.

This, in a nutshell, is the central challenge to a Walker candidacy: Can a strategy tailored to winning Iowa allow him to survive beyond that?

\"I think what he's trying to do is to find his voice and to try to find his voice nationally,\" said Doug Heye, a GOP strategist and campaign veteran. \"That's a difficult thing for somebody who's only campaigned in one state his whole career.\"

Walker will make his candidacy for President official on Monday and then embark on a tour of the early voting states, culminating in a three-day jaunt across Iowa in an RV.

Seizing the Hawkeye State

Privately, Walker aides and allies admit that Iowa is their top prize. Recent polls show Walker with an average lead of nearly eight points in the state, according to Real Clear Politics. For a candidate who is casting himself as \"fighter\" and touting three election victories in four years, the Hawkeye State is a must-win.

Walker's advisers dismissed the notion that there's been any change in his approach on key issues.

\"There is no story behind the story,\" Rick Wiley, one of Walker's top advisers, said of his approach to the same-sex marriage decision. \"He always thought that this should be a decision left up to the states.\"

But Walker's shifting attempts to strike the right balance on lightning rod issues from immigration to same-sex marriage to abortion highlight the conundrum he faces in trying to build a winning coalition in socially conservative Iowa and remain viable in the states beyond, donors and GOP strategists said.

Charlie Sykes, a Wisconsin conservative radio host who has known Walker well for two decades, said Walker's strident tone on the gay marriage ruling last month caught people back home off guard because historically he had been \"almost agnostic\" on the subject.

\"I was surprised, because I thought there was probably an easier way to finesse that,\" said Sykes.

When a federal court struck down a same-sex marriage ban in Wisconsin last year, Walker said the fight was over in Wisconsin and deemed it an issue for officials at the federal level. Even then he didn't call for a constitutional amendment to ban same-sex marriage.

\"I think it's resolved,\" Walker told reporters at the time.

In Wisconsin, Sykes explained, Walker didn't have to navigate a fractured GOP, which is at the heart of the Republican presidential primary process.

\"I don't think that Scott wants to be outflanked on the right,\" said Sykes.

On immigration, Walker has moved from his support for a pathway to citizenship for undocumented immigrants to declaring he is not in favor of amnesty and, at times, even sounding protectionist as he questioned the country's approach to legal immigration.

His packaging of his anti-abortion views has been through a similar evolution. In an ad for his 2014 reelection race, Walker touted his pro-life position, but went on to say he supported legislation that \"leaves the final decision to a woman and her doctor.\"

Now Walker is taking a harder line, preparing to sign a bill in Wisconsin that would ban all abortions after 20 weeks, including in the case of rape and incest.

The approach appears to be currying favor for Walker among some social conservatives. Penny Young Nance, chief executive of the conservative group Concerned Women for America, said she's grown more comfortable with Walker's conservative credentials after spending time with him on a couple of occasions.

\"We're starting to see who he is,\" she said. \"I think he's being very plainspoken and gone the extra mile to be very clear on what he believes.\"

Others still need some convincing. Bob Vander Plaats, chief executive of the Iowa social conservative group The Family Leader, describes his relationship with Walker as a courtship in the early stages.

\"It's not that we don't believe him,\" said Vander Plaats, who will host Walker and other GOP candidates at a forum later this month. \"We have to kind of make sure that that's a genuine belief.\"

A challenge for Republicans

Taking a sharp turn to the right in order to win the top spot in Iowa and other GOP primaries is hardly a new challenge for Republicans. Social conservative firebrands Mike Huckabee, the former governor of Arkansas, and Rick Santorum, a former Pennsylvania senator, both notched victories in Iowa but failed to win broader support to clinch the nomination.

Mitt Romney's conservative shifts in 2012 gave Democrats plenty of ammunition to paint him as a flip-flopper. In an effort to bolster his credentials with the right, Romney called for \"self-deportation\" for undocumented immigrants, a comment that haunted him in the general election and helped President Barack Obama sail to victory.

The far-right allure is a particular concern for Republican donors, who are tired of seeing their candidates bogged down in battles over social issues and crippling themselves with Hispanic voters. On some issues, such as gay marriage, many donors are more progressive. A number of prominent GOP donors support same-sex marriage.

At a recent fundraising retreat in Utah, hosted by Romney, one donor asked Walker how he would avoid being dragged into a prolonged fight over social issues. Walker said it was important to be disciplined and cognizant that a national presidential race is broader than a gubernatorial campaign in Wisconsin.

\"I'm a social conservative, as well as economic and fiscal one, so I don't think that the answer is to run away from your position,\" Walker said. \"Just make it clear to the American people that this is where you stand.\"

While voters might be familiar with the governor's election victories or his fight with the unions, his campaign plans to flesh out his record in Wisconsin, particularly on education and economic issues, to make their case. His advisers are already looking further down the field to opportunities in states that vote in early March, like Tennessee, North Carolina, Georgia and Alabama that could help him win the nomination.

Walker's team hopes to prove his message resonates across party lines, Wiley said. One point in their favor: When Walker won re-election in Wisconsin in 2014, he carried independent voters by 11 points, exit polls show.

\"You can see the true benefit that a family in Wisconsin has received over the last four years,\" said Wiley. \"That's the great untold story. Everybody just thinks this guy took on the unions and he won, but there's a lot more to it.\"

In fact, Walker's early strength in Iowa shows that his message is already making inroads with different factions, from social conservatives to fiscal-oriented voters, said Matt Strawn, former chairman of the Iowa Republican Party.

\"The good news for Walker is he's demonstrating support across disparate groups of Iowa Republicans,\" Strawn said. \"The downside is that means the shots are going to be coming from all sides.\"

CNN's Dana Bash and Maeve Reston contributed to this report.