Investigators are piecing together the timeline of Loughner's frenzied morning before the attack that killed six.
"What he did and the morning before the shooting, we're just trying to find all that out," said sheriff's Capt. Chris Nanos.
Nanos wouldn't say what Loughner bought during two trips to Walmart.
After the shopping trips, Loughner ran a red light but was let off with a warning, the Arizona Game and Fish Department said. The officer took Loughner's driver's license and vehicle registration information at 7:30 a.m. but found no outstanding warrants and didn't search the car, a late 1960s dark gray Chevy Nova.
About 8 a.m., Randy Loughner saw his son walk to one of the family's vehicles and take a black bag out of the trunk.
"The father went out and said, 'What's that?' and he mumbled something and took off running," said Sheriff Clarence Dupnik.
Randy Loughner got in his truck and chased his son, but Jared ran into the desert.
At 10:11 a.m., police say Jared Loughner showed up at a Tucson grocery store in a taxi and shot 19 people, killing a federal judge and a 9-year-old girl, among others and wounding more than a dozen.
Hours after the attack, sheriff's deputies swarmed the Loughners' home and removed what they describe as evidence Loughner was targeting Giffords. Among the handwritten notes was one with the words "Die, bitch," which authorities believe was a reference to Giffords.
Investigators with the Pima County Sheriff's Department previously said they found handwritten notes in Loughner's safe reading "I planned ahead," "My assassination" and "Giffords." Capt. Chris Nanos said all the writings were either in an envelope or on a form letter Giffords' office sent him in 2007 after he signed in at one of her "Congress on Your Corner" events -- the same kind of gathering where the massacre occurred.
Meanwhile, the city held a tribute to victims the eve of a presidential visit.
On Tuesday night, several hundred mourners filled a Tucson church for a public Mass to remember the slain and pray for the injured. As people filed in, nine young girls sang "Amazing Grace." The youngest victim of the attack, 9-year-old Christina Taylor Green, was a member of that choir.
"I know she is singing with us tonight," said Tucson Bishop Gerald Kicanas, who presided over the service.
President Barack Obama visits Arizona Wednesday and will honor the victims in a speech to a rattled state and nation.
In one apparent reaction to the shooting, the FBI said background checks for handgun sales jumped in Arizona following the shootings, though the agency cautioned that the number of checks doesn't equate to the number of handguns sold.
Still, there were 263 background checks in Arizona on Monday, up from 164 for the same day a year ago -- a 60 percent rise. Nationally, the increase was more modest: from 7,522 last year to 7,906 Monday, a 5 percent jump.
Loughner's parents, silent and holed up in their home since attack, issued a statement Tuesday, expressing remorse over the shooting.
"There are no words that can possibly express how we feel," Randy and Amy Loughner wrote in a statement handed to reporters waiting outside their house. "We wish that there were, so we could make you feel better. We don't understand why this happened.
"We care very deeply about the victims and their families. We are so very sorry for their loss."
Four days after being shot in the head, Giffords is making small movements on her own, tugging at her hospital gown and touching her wounds, one of her doctors said Wednesday.
"We have really decreased the amount of sedation we are giving her and as a result of that, she's becoming more and more spontaneous all the time," said Dr. Peter Rhee, trauma chief at the University of Arizona.
In addition to the new details about the hours before the shooting, interviews with those who knew Loughner or his family painted a picture of a young loner who tried to fit in.
Before everything fell apart, he went through the motions as many young men do nowadays: Living at home with his parents, working low-wage jobs at big brand stores and volunteering time doing things he liked.
None of it worked. His relationship with his parents was strained. He clashed with co-workers and police. And he couldn't follow the rules at an animal shelter where he spent some time.
Loughner grew up on an unremarkable Tucson block of low-slung homes with palm trees and cactus gardens out front. Fittingly, it's called Soledad Avenue -- Spanish for solitude.
Solitude found Loughner, even when he tried to escape it. He had buddies but always fell out of touch, typically severing the friendship with a text message. Zach Osler was one such friend.
Loughner's father moved into the house as a bachelor, and eventually got married, longtime next-door neighbor George Gayan said. Property records show Randy Loughner has lived there since 1977. Unlike other homes on the block, the Loughners' is obscured by plants. It was assessed in 2010 at $137,842.
Randy Loughner apparently has not worked for years -- at least outside his home.
Amy Loughner got a job with the county parks and recreation department just before Jared was born, and since at least 2002 has been the supervisor for Roy P. Drachman Agua Caliente Park on the outskirts of the city. She earns $25.70 an hour, according to Gwyn Hatcher, Pima County's human resources director.
Associated Press writers Alicia Chang and Gillian Flaccus in Tucson, Jacques Billeaud and Bob Christie in Phoenix, Christy Lemire in Los Angeles and news researcher Julie Reed in Charlotte, N.C., contributed to this report.