As Gov. Kathy Hochul sampled a sandwich at the New York State Fair on Sunday, touring the Syracuse-area spectacle like other governors before her, she overtly embraced her role as the state’s new leader — and implicitly set down a marker for 2022, when she intends to seek election to a full term as governor.

Two days later, the New York City public advocate, Jumaane D. Williams, was in town, observing the cows and swinging by a butterfly garden. On Wednesday, it was Attorney General Letitia James’s turn.

Ms. James greeted attendees, admired a butter sculpture and, like Mr. Williams, stoked fresh speculation about future political ambitions — and whether those ambitions included a run for governor.

All three New York Democratic officials have visited the fair before. But the pilgrimages this week — not unlike a presidential hopeful’s early visits to Iowa — took on fresh resonance, offering a very public reminder of a nascent political contest that has been brewing behind the scenes.

After more than a decade of governors’ races that were dominated and defined by the now-disgraced former Gov. Andrew M. Cuomo, New York’s political class is quietly beginning to plan for a different — and possibly, fiercely contested — primary campaign next year.

While Ms. Hochul, New York’s first female governor, has been in office for just over a week, the primary machinery is already whirring to life, with hiring, polling and political gamesmanship picking up speed. One poll had extensive questions about Mayor Bill de Blasio, who has not ruled out running for governor.

The most concrete activity among Democrats surrounds Ms. Hochul. She has already brought on strategists with national and New York experience; Tucker Green, a major Democratic fund-raiser, has also recently joined her campaign team as Ms. Hochul works to cement fund-raising strength, often an advantage for a sitting governor.

The governor is making other decisions about her campaign infrastructure and will have more personnel announcements after Labor Day, an adviser to Ms. Hochul said.

Many New York Democrats expect Ms. Hochul to be a powerful contender, boosted by the advantages of incumbency, the statewide network she has already assembled and an outpouring of good will for a new governor who has moved urgently to restore some of the norms and relationships that crumbled under the previous chief executive.

But the field will also be shaped by Ms. Hochul’s track record as she navigates a series of staggering challenges facing the state.

“We don’t know who is going to be in it,” said Representative Nydia M. Velázquez, a New York Democrat. “Who is going to be in it will be defined by Kathy Hochul’s leadership.”

On Wednesday, Ms. Hochul announced that Kathryn Garcia — a former mayoral candidate who had been mentioned as a possible candidate for governor herself — had been appointed director of state operations.

Perhaps the biggest uncertainty in the race is whether Ms. James will run. Some of her advisers, including some at the Hamilton Campaign Network, which was heavily involved in Ms. James’s previous runs, are beginning to have conversations about who could join a potential James bid for governor, according to people familiar with the discussions — part of an effort among Ms. James’s allies to keep her options open.

“We do not comment about our clients,” the company said in a statement.

“Tish would be an excellent governor,” said John Samuelsen, international president of the Transport Workers Union, who lauded her “courageousness” in spearheading the investigation that led to Mr. Cuomo’s resignation. “She has a demonstrated record of steadfast support for working people.”

People who have spoken with Ms. James in recent weeks have not gotten the impression that she has made a decision, but one of those people said Ms. James indicated she would like to make a call sometime this fall, ahead of the Democratic State Convention slated for early next year. Others have the broad sense that she would be inclined to see how Ms. Hochul’s early months as governor proceed.

Asked about those discussions, a representative for Ms. James said that the attorney general is “fully focused on her work protecting and defending the rights of New Yorkers and plans to continue taking on the big fights that matter.”

There has been a flurry of activity in other potential candidates’ camps, too.

Recently, Anna Greenberg, Mayor de Blasio’s longtime pollster, conducted a survey testing the mayor’s appeal outside of New York City and the potency of particular messages about him.

One Westchester resident, who took notes as he was polled on Tuesday, said that questions tested the appeal of several potential candidates for governor, including Ms. Hochul, Mr. Williams, Ms. James and Mr. de Blasio.

Then, several specific messages about Mr. de Blasio — questions that were not raised about the other potential candidates — were tested.

Among other things, the questioner discussed Mr. de Blasio’s record of battling Mr. Cuomo over his response to the pandemic, his efforts to provide legal services to New Yorkers facing evictions and his work on police reform and universal prekindergarten. Then the pollster asked if those facts made the respondent more or less inclined to support him.

A spokesman for Mr. de Blasio declined to comment.

On Long Island, Steven Bellone, the Suffolk County executive, has hired J.J. Balaban and Brandon L. Davis, veteran Democratic political strategists and ad makers, and brought on the national firm GPS Impact as he contemplates a run for governor.

Representative Thomas Suozzi, a Long Island Democrat, is also thought to be seriously considering a run. But he intends to assess how Ms. Hochul performs and wants to accomplish his goals in negotiations in Congress over the federal deduction for state and local taxes, according to one Long Island Democrat with knowledge of Mr. Suozzi’s intentions, granted anonymity to discuss private deliberations.

In recent weeks, Mr. Williams has said publicly that he is exploring a bid for governor, and privately he has told at least one person that he has already decided to run — withholding his plans because he does not want to announce it so close to Ms. Hochul’s swearing-in.

In a brief interview, Mr. Williams said he thinks it is important to give the sitting governor time “to get her bearings,” and for the state to “take a moment to recognize the historic nature of the first woman governor.”

“There’s definitely time to have those conversations in the near future,” Mr. Williams said. “And I have said I am considering, but it is important that we allow that time period before we dive deep into those questions.”

Jay S. Jacobs, the chairman of the New York State Democratic Party, suggested that there could be risks for candidates announcing any intentions so soon after Ms. Hochul assumed the governorship.

“Right now it would be premature and probably unseemly,” he said. “You have to have a reason, and I think that means you have to give the current governor a little bit of time. Then you can distinguish yourself from her, if you choose to run.”

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