When the Supreme Court returns to the bench on Monday, it will face a docket filled with unfinished business. The justices will revisit issues like gun rights, government power, race and free speech even as they are shadowed by intense scrutiny of their conduct off the bench.

In the coming months, moreover, the court will very likely agree to hear a major abortion case, one that could severely limit the availability of a drug used in more than half of all pregnancy terminations. A decision in that case could come in June, two years after the court overturned Roe v. Wade.

Recent history suggests that the court’s six Republican appointees will continue to move the law to the right. The main questions are how far, how fast and what impact the questions swirling around the justices’ ethical standards will have on their judicial work and personal relationships.

It has been almost three years since Justice Amy Coney Barrett joined the court, creating a six-justice conservative supermajority. In her first full term, ending in June last year, that majority ran the table in blockbuster cases on abortion, guns, the environment and religion.

The most recent term, which ended this summer, also featured significant conservative victories, on affirmative action, student loans and gay rights. But the balance of the court’s decisions presented a more complicated picture, featuring occasional liberal victories from unusual alliances.

“The question for this term is whether last term’s voting behavior will hold or whether the court will snap back to a 6-3 court,” said Irv Gornstein, the executive director of Georgetown’s Supreme Court Institute.

The new term will test Chief Justice John G. Roberts Jr.’s ability to steer his court toward the sort of incrementalism he has long espoused, perhaps in alliances with Justices Barrett and Brett M. Kavanaugh. If liberals achieve some victories, though, those may come because litigants and lower courts had staked out positions too extreme for even a fundamentally conservative Supreme Court.

A Second Amendment case, to be argued in November, could be an example of such potential overreach. The case, United States v. Rahimi, No. 22-915, is an appeal from a decision striking down a federal law that disarms people subject to domestic violence restraining orders.

The appeals court, relying on a transformative Supreme Court ruling last year that required the government to identify historical analogues to justify laws limiting gun rights, said there was no comparable historical practice.

The Supreme Court’s ruling will have sweeping consequences, said Joseph Blocher, a law professor at Duke.

“Courts have really struggled to find meaningful historical guidance about the constitutionality of modern gun laws like those regulating high-capacity magazines or guns on the subway,” he said. “Rahimi gives the justices a chance to clarify that analogizing from history isn’t a recipe for anachronism.”

The coming term could also take large strides toward achieving a long-sought goal of the conservative legal movement: stripping administrative agencies of the power to regulate.

“This term’s administrative law cases present an existential question,” said Kate Shaw, a professor at the Benjamin N. Cardozo School of Law. “Will our government retain the power and capacity to respond to the most pressing issues of our time? Or will the court continue to incapacitate the administrative state, limiting its authority and substituting the court’s own views and preferences for those of expert agencies?”

One case asks the justices to overturn a major precedent, Chevron v. Natural Resources Defense Council, which requires courts to defer to agencies’ reasonable interpretations of ambiguous statutes. The decision shifted power from Congress and the courts to agencies, and it is unpopular among businesses subject to many kinds of regulations, including of the environment, the workplace and the marketplace.

The new case, Loper Bright Enterprises v. Raimondo, No. 22-451, arose from a federal law that allows the National Marine Fisheries Service to require fishing vessels to carry federal monitors to prevent overfishing. The service interpreted the law to let it order the fishing industry to pay the monitors’ salaries, a position an appeals court found reasonable.

In the past two terms, the court has discarded precedents on abortion and affirmative action. “Overruling Chevron,” the Biden administration told the justices in a recent brief, “would be a convulsive shock to the legal system.”

A second administrative law case, this one concerning the Consumer Financial Protection Bureau, could hobble that agency, imperil every regulation and enforcement action it took in the 12 years of its existence and cast a shadow over the constitutionality of other government entities, not least the Federal Reserve.

The central question in the case, Consumer Financial Protection Bureau v. Community Financial Services Association of America, No. 22-448, is whether the way Congress chose to fund the agency violated the Appropriations Clause of the Constitution, which says that “no money shall be drawn from the Treasury, but in consequence of appropriations made by law.”

The bureau is funded by the Federal Reserve System, in an amount determined by the bureau so long as it does not exceed 12 percent of the system’s operating expenses.

The third administrative law case, Securities and Exchange Commission v. Jarkesy, No. 22-859, is a challenge to enforcement actions in securities fraud suits heard by S.E.C. judges. A federal appeals court ruled that those actions violated the defendants’ constitutional right to a jury trial and must instead be brought in federal court.

Many agencies use similar administrative proceedings, so a ruling against the S.E.C. could have a broad impact.

The three cases will provide powerful clues about the fate of executive power, said Jonathan H. Adler, a law professor at Case Western Reserve University.

“After a lot of hints and feints,” he said, “this term will likely reveal the scope of the justices’ ambitions with regard to the administrative state.”

The term may also illuminate how far the court is willing to go in curtailing access to abortion. After the court eliminated the constitutional right to abortion, a coalition of anti-abortion doctors and organizations challenged the Food and Drug Administration’s approval more than two decades ago of an abortion pill.

At stake is the availability of the pill, mifepristone, which more than five million women in the United States have used to terminate their pregnancies, though the outcome could also undermine the agency’s authority to approve and regulate other drugs.

In the term that ended in June, a closely divided court, in a surprise decision, ruled that Alabama had diluted the power of Black voters in drawing a congressional voting map, reaffirming part of a landmark civil rights law that had been thought to be in danger. On Oct. 11, the court will turn to a related question: whether a congressional voting district in South Carolina should be restored after a lower court struck it down as an unconstitutional racial gerrymander.

Last term, the court largely sidestepped cases concerning what responsibility tech companies should have for what their users post on their sites.

In the new term, the justices will again confront the intersection of free speech and technology, including one case that sets up a major showdown over the First Amendment. On Friday, the court agreed to hear appeals on whether the Constitution allows Florida and Texas to prevent large social media companies from removing posts based on the views they express. And this month, the court will hear arguments on whether elected officials violated the First Amendment when they blocked people from their social media accounts.

Not all of the cases in the coming term give rise to a sense of déjà vu. The court will, for instance, weigh in on a bankruptcy settlement involving Purdue Pharma that would give billions of dollars to those harmed by the opioid epidemic but shield members of the Sackler family, which once owned the company.

In August, the court temporarily blocked the deal, delaying any payment to thousands of plaintiffs who sued the family and Purdue Pharma, which makes the prescription painkiller OxyContin. The court will hear arguments in the case, Harrington v. Purdue Pharma, No. 23-124, in December.

The court will do its work as calls mount for it to address what critics say were grave ethical lapses by the justices.

In particular, Justice Thomas for decades accepted gifts and trips from Harlan Crow, a Texas billionaire who has donated to conservative causes, and did not disclose them. Mr. Crow’s largess extended to covering private school tuition for the grandnephew Justice Thomas was raising and buying the home of the justice’s mother. Justice Samuel A. Alito Jr., for his part, flew on the private jet of a hedge fund magnate with business before the court and did not recuse himself in related cases.

The revelations renewed calls for the justices to adopt an ethics code, and while Chief Justice Roberts and Justices Kavanaugh and Elena Kagan have all said that the court was at work on a solution, nothing concrete has emerged.