AP) — The family of a former U.S. Marine imprisoned in Iran since his arrest three years ago on spying charges said his conditions have improved and he is now allowed to call home several times a week.

The sister and brother-in-law of Amir Hekmati acknowledge that's a big step forward for a man who spent his first 16 months held in solitary confinement in Iran's notorious Evin prison, north of the capital, Tehran. But Hekmati's relatives are unwavering in their goal to obtain his release.

"The important thing here is that he needs to come home. And our push is one of urgency," Hekmati's brother-in-law, Ramy Kurdi, told The Associated Press in an interview. Hekmati's father is dying of cancer, and "every day matters. Every day is an injustice," Kurdi said.

Hekmati, a dual U.S.-Iranian citizen born in Arizona and raised in Michigan, was arrested in August 2011, then tried, convicted and sentenced to death for spying. Hekmati appealed, and Iran's Supreme Court annulled the death sentence and ordered a retrial in 2012. The country's Revolutionary Court then overturned his conviction for espionage, instead charging him with "cooperating with hostile governments" and sentenced him to 10 years in prison.

The U.S. government repeatedly has denied that Hekmati is a spy. His family, which lives in Flint, Michigan, says he is innocent and only went to Iran to visit his grandmothers.

Kurdi and his wife, Hekmati's sister, Sarah, were in New York to press the case for their brother as world leaders, including Iranian President Hasan Rouhani, meet in New York for the start of the annual U.N. General Assembly. The meetings also coincide with another round of delicate negotiations focused on reducing Iran's atomic activities in exchange for an end to nuclear-related sanctions.

Mahmoud Alizadeh Tabatabaei, Hekmati's lawyer in Iran, has said he would appeal his client's 10-year prison sentence. Kurdi said one of the many options being considered was seeking a pardon for Hekmati.

"We're looking for any release. It doesn't matter to us what the medium is, what matters is just the outcome," Kurdi said.

"He could be released on the basis of the fact he's been a model prisoner, and has complied with the rules. He could be released on a humanitarian basis. The conviction itself could be set aside. There are a number of ways that he could be released," said U.S. Rep. Dan Kildee, a Michigan Democrat whose district includes Flint and who accompanied Hekmati's relatives to New York

The regular phone calls back home, which began about a month ago, have helped the family and Hekmati handle their grief. Hekmati is able to purchase calling cards through his prison account, and his mother arranges to have funds put on the cards for him, Sarah Hekmati said.

"Emotionally it's helped significantly just to hear his voice, even just to lift his spirits with day-to-day stories about his niece and nephew, how my dad's rehab is going, any updates about his life that he's left behind here," Sarah Hekmati said. "There are things that we're trying to manage for him and keep afloat so that he hopefully has a life to come back to when and hopefully, God willing, he is released."

But the calls, which Sarah Hekmati said generally last five to seven minutes, also are a painful reminder of their fruitless efforts to bring Hekmati home. And their father's illness — he was diagnosed with brain cancer two years ago and has recently also had a stroke — adds to the general stress.

"It's very devastating for my brother because he feels helpless ... and he's always fearful that he's going to get the news of something tragic happening to his dad while he's in prison," Sarah Hekmati said. "One of his biggest concerns when he does call the family is how dad is doing. And he's giving us pep talks to keep us strong while he's in prison."