The Arizona Democrat kept the door open when asked about such a possibility, saying "well, a little later" and "maybe" during an interview on NBC's "Today" show.
The interview was aired a day after bell-ringing and flag-raising ceremonies were held in Tucson to remember the six people killed and 13 injured in the Jan. 8, 2011, shooting that occurred as Giffords met with constituents outside a grocery store.
To commemorate the anniversary and how far she has come, Giffords jumped from an airplane in Arizona in a tandem skydive.
Giffords, who did the "Today" interview with husband Mark Kelly at her side, said she felt "happy and sad" on the anniversary but believes it's time to move on.
She said she's making slow progress in her rehabilitation and wants to work on her Spanish and resume playing the French horn.
The interview was conducted by "Today" co-host Savannah Guthrie, who said a person associated with Giffords had contacted NBC several weeks ago about Giffords' intention to make the parachute jump that was broadcast on the NBC show.
Broadcast video showed Giffords and other jumpers step out the back of a twin-engine plane and join hands midair during their descent under a blue sky.
Giffords landed on a grass lawn without injury. She waved and blew kisses to onlookers who cheered and applauded.
"She was the least nervous person on the plane," said Jimmy Hatch, a former Navy SEAL who accompanied the group. "They did a little moment of silence at the drop zone ... The emotion was really heavy. Then she smiled and said, 'Let's go.'"
Giffords has become a leader of Americans for Responsible Solutions, a national organization she founded with her husband to rival the powerful pro-gun lobby.
The group has struggled to bring about major change in its first year of existence, but the couple is confident they've laid the groundwork for success in future election cycles.
Giffords and Kelly formed their organization just weeks after the school massacre in Newtown, Conn.
Since then, Congress has done nothing to tighten the nation's gun laws.
Some states, including Colorado and Delaware, pushed ahead with their own gun-control measures, while others moved in the opposite direction. Arizona, the home state of Giffords, passed a law that requires municipalities to sell weapons surrendered at buyback programs instead of destroying them.
Kelly said his group raised more than $11 million between January 2013 and July 2013.
"So we're going to have the resources to be effective in the next election cycle in 2014," he said.
In an opinion piece for The New York Times on Wednesday, Giffords wrote about her struggles to recover, calling it "gritty, painful, frustrating work."
"I had planned to spend my 40s continuing my public service and starting a family," she wrote. "Instead, I've spent the past three years learning how to talk again, how to walk again."
Jared Lee Loughner was sentenced in November 2012 to seven consecutive life sentences, plus 140 years, after he pleaded guilty to 19 federal charges in the shooting.
In Tucson, about 100 residents attended a ceremony on Wednesday outside the University of Arizona Medical Center, where the injured were treated.
Other ceremonies and moments of silence took place across the city.
"I think the commemorations are, in large part, recognition of our community's collective care and compassion and grit to go on," Tucson Mayor Jonathan Rothschild said.
Pam Simon, 66, a Giffords aide at the time who was shot in the chest, reflected with crisp memories and a positive outlook.
"When we stop on an anniversary to really reflect, sometimes it opens the wounds a little bit," she said. "But it's also gratifying in a way to see the community come out again and remember."
Tang reported from Phoenix. Paul Davenport contributed from Phoenix.