Smith was shot in the back and the side, according to a warrant read in court Sunday night before a magistrate set bail at $1 million for Hayes.
Police said Smith, 34, was killed in a case of road rage by a man who had rear-ended his car. They arrested Hayes, 28, on a charge of second-degree murder. Prosecutors have 60 days to decide whether to bring that charge against Hayes.
Officials said Hayes' Humvee H2 rear-ended Smith's Mercedes G63 around 11:30 p.m. Saturday in the upscale Lower Garden District, pushing it into a Chevrolet Impala driven by two of Smith's acquaintances. The two argued, and Hayes shot Smith and Smith's wife, Racquel, police said. She was taken to a hospital with a leg wound.
Defense attorney John Fuller asked for low bail Sunday night, noting that Hayes had waited for police, owns a towing business and is raising his 5-year-old son.
Magistrate Brigid Collins said she understood, but, according to the warrant, "I've also got a gentleman who was shot in the back."
Smith and his wife, who have three children, had been at the French Quarter Festival, which features local music and food, in the hours before the shooting. He posted a photo of himself and his wife on Twitter and Instagram.
Hayes and Smith didn't know each other, Fuller said. But, he said outside the courthouse, the accident that police said touched off the deadly confrontation was the second within moments.
He said Hayes ran into the back of Smith's Mercedes while following a hit-and-run driver who had rear-ended his Humvee and sped off. Fuller wouldn't say whether Hayes identified the Mercedes as that vehicle.
Hayes is not guilty of murder, and the story is more complex than the police account, Fuller said after the hearing. He said Hayes called 911 after he was hit and persuaded a witness who was about to leave to remain and talk to police.
Police spokesman Tyler Gamble told the Associated Press in an email Sunday that he had no immediate information about the witness or the 911 call.
Online court records show Hayes pleaded guilty in 2014 to one count each of possessing an illegal weapon and possessing drug paraphernalia.
The gun he had was legal, Fuller said, noting that police didn't accuse him of any weapons charge. Gamble said the gun hadn't been reported stolen, but tracing its actual history takes more time.
Hayes sued the New Orleans Police Department and six officers after police killed his father in 2005. Police settled the lawsuit in 2011. The settlement is confidential, said attorney Ike Spears, who represented Hayes in that lawsuit.
Smith was beloved by fans for his role in New Orleans' 2009 Super Bowl championship, though the defensive captain also was at the center of the NFL's bounty probe in 2012, the last year he played in an NFL regular-season game.
The league concluded that Smith and fellow defensive captain Jonathan Vilma helped run a locker-room pool that paid cash bonuses for heavy and even injurious hits.
Smith successfully appealed a four-game suspension, though coach Sean Payton was suspended for the entire 2012 season.
A year earlier, however, Smith served a two-game suspension for using a weight-loss product that included a diuretic banned by the league for masking steroid use.
Smith, a native of Queens, N.Y., played for Ohio State's 2002 national championship team and graduated in 2005 with a degree in criminology.
New Orleans chose him in the first round of the 2004 draft. He led the Saints with a career-high 13 sacks in 2009. Smith's 67 1/2 career sacks rank fourth in Saints history. The New Orleans Saints Hall of Fame says Smith had been unanimously voted into the hall last month.
Off the field, Smith established a foundation "to motivate, educate and provide opportunities for women and children."
The Smith family said in a statement Sunday that they were thankful for the outpouring of support but requested privacy as they grieve.