In the year since a one-time window opened in New York State allowing people to file sex-abuse lawsuits even after the statute of limitations had expired, more than 3,000 civil suits have been filed.

Before the deadline on Thanksgiving, a flurry of attention-grabbing suits were filed against politicians — like former Gov. Andrew Cuomo and Eric Adams, the mayor of New York — and celebrities, like Sean Combs, the producer and music mogul, who had just settled a separate suit filed in Federal District Court in Manhattan accusing him of rape.

But hundreds of people have also — collectively and separately — sued institutions, including the state’s prisons, jails and prominent hospitals, for abuses they said were systematically ignored and hidden for decades. At least 479 suits contain charges of abuse at Rikers Island.

As the legislation to allow the civil suits, known as the Adult Survivors Act, approached its sunset date, the number of lawsuits filed — both in State Supreme Court and in the Court of Claims — steadily increased after a campaign to alert people to the deadline. The number of cases filed in State Supreme Court alone rose from 803 on Oct. 31 to 1,397 as of Nov. 22.

“The mission of the Adult Survivors Act, allowing survivors of sexual abuse the opportunity to speak their truth in court and seek redress, has been realized,” said Linda Rosenthal, a state assemblywoman on the Upper West Side who was one of the bill’s sponsors.

Now, the focus is shifting to new legislation that could create another window for lawsuits or make permanent the ability to file civil lawsuits for sexual assault beyond the statute of limitations in New York State, she said.

In 2019, New York expanded the statute of limitations to 20 years for both civil and criminal cases involving some categories of sex crimes, like rape. Before, the statute was generally just one year for civil cases. However, the expanded statue only covered cases going forward, not retroactively.

The Adult Survivors Act, which passed the State Assembly in May 2022, was created to enable victims who were 18 or older at the time of the alleged abuse a one-time opportunity to file civil lawsuits against individual people or institutions, even when the statute of limitations had run out. They had a year, starting with Thanksgiving 2022.

The law followed the state’s Child Victims Act, which included a one-year “look-back window” during which old childhood sex-abuse claims whose statute of limitations had expired could be revived.

One law firm, Slater Slater Schulman LLP, said it had filed 1,218 cases against New York State, 479 cases against New York City and 74 cases against counties across the state for decades-long allegations of sexual abuse at the hands of corrections officers.

Although a one-year window was a good step, it did not provide enough time to capture everyone who could have a civil claim, said Mallory C. Allen, a partner at PCVA Law who represents clients who have sued through the Adult Survivors Act.

Ms. Allen’s firm has filed more than 100 lawsuits against a urologist, Darius A. Paduch, who was charged by federal prosecutors in Manhattan in April with carrying out yearslong abuse against minor patients. Those who have filed under the Adult Survivors Act against Mr. Paduch and the prominent New York hospitals where he worked, primarily NewYork-Presbyterian/Weill Cornell Medical Center, were adults when they say they were abused.

“He was arrested, and the news broke of his arrest during the window,” Ms. Allen said, adding: “I’m confident that we will continue to receive phone calls from survivors who want to exercise their legal rights.”

The first filing against Dr. Paduch was in December last year, Ms. Allen said, meaning many of his former patients may not have heard about the case or haven’t had enough time to process the news and then file a lawsuit.

Just last month, 19 suits were filed against Dr. Paduch. “The truth must be brought into the light, for every victim of every age,” David Sonneborn, one of the plaintiffs, said in a statement at the time.

Columbia University, whose hospital system is facing hundreds of suits through the law, responded to pressure several weeks ago and said it would notify nearly 6,500 patients of a former gynecologist at the hospital who was sentenced to 20 years in prison for abusing patients. The former doctor, Robert A. Hadden, was first arrested in 2012.

It can take years before many victims of abuse are ready to come forward with their accusations, said Michael Polenberg, vice president of government affairs at Safe Horizon, a victims’ services group, which pushed for the legislation.

“These things can take time,” he said. “And that is not an aberration — that is actually very typical of survivors’ experience.”

Extending the filing window, or implementing a permanent solution, will require new action by the State Legislature, said State Senator Brad Hoylman-Sigal, who sponsored the bill in the upper chamber.

“I’m hopeful that I can convince colleagues of either another extension or permanent window,” Mr. Hoylman-Sigal said. Some jurisdictions are already looking to open permanent filing windows, particularly for victims of childhood abuse, he said.

Will Barclay, the Republican minority leader in the Assembly, said that since Republicans had supported the original bill, which brought “alarming cases to light and provided victims with a path to justice,” they would consider an expanded window.

But statutes of limitations “exist for a reason,” Mr. Barclay added. “Unilaterally erasing the threshold would necessitate a deeper discussion, ideally with all stakeholders within the justice system.”

Liz Roberts, Safe Horizon’s chief executive, said she hoped that the lawsuits filed this year — and how many have targeted prisons, hospitals and other large institutions — would result in changes to how reports of sexual abuse are handled.

“I hope and pray it will be a watershed moment for us as a country,” she said.