Recent CT scans showed no further swelling in the brain, but doctors were guarded. Typically, the third day represents a "landmark" in terms of monitoring swelling, but neurosurgeon Dr. Michael Lemole cautioned that Tuesday "does not have to be that day."
"We might have to be on pins and needles," for a few more days, Lemole told "The Early Show" Tuesday.
On Monday, Lemole said that while Giffords wasn't "out of the woods yet," but the more time goes by without an increase in swelling, "we're slightly more optimistic."
After Saturday's operation to temporarily remove half of her skull, doctors over the past two days had Giffords removed from her sedation and then asked basic commands such as: "Show me two fingers."
"When she did that, we were having a party in there," said Dr. Peter Rhee, adding that Giffords has also been reaching for her breathing tube, even while sedated.
"That's a purposeful movement. That's a great thing. She's always grabbing for the tube," he said.
Lemole said Tuesday that until Giffords' breathing tube is removed, doctors will be limited to those simple neurological tests.
"We're not there yet," he said.
Giffords' family is by her side, receiving constant updates from doctors. On Monday, two well-known doctors with extensive experience in traumatic brain injury were traveling to Tucson to help consult on Giffords' case.
Her doctors have declined to speculate on what specific disabilities the 40-year-old congresswoman may face.
Lemole said that recoveries from brain injuries are usually measured in weeks and months, not days and there is "no reason to expect different here."
Two patients injured in the shooting were discharged from the Tucson hospital Sunday night. Seven others remained hospitalized.
The lone suspect in the attack, Jared Loughner, was orderedat a Monday court appearance. Law clerks, courtroom artists and reporters alike turned from the wooden benches to look at the vacant-eyed 22-year-old, accused of trying to assassinate Giffords and killing a federal judge along with five others.
Wearing a beige prison jumpsuit and handcuffs and sporting a pink gash on the hairline of his shaved head, Loughner on Monday afternoon spoke just a brief reply when the judge asked if he understood that he could get life in prison - or the death penalty - for killing federal Judge John Roll.
"Yes," he said.
With few new details emerging at Monday's hearing, questions remained about what could have motivated someone to arm himself with a pistol and magazines carrying 33 bullets each, and rain gunfire on a supermarket parking lot crowded with men, women and children.
And who exactly was Jared Loughner?
Comments from friends and former classmates bolstered by Loughner's own Internet postings have painted a picture of a social outcast with almost indecipherable beliefs steeped in mistrust and paranoia.
"If you call me a terrorist then the argument to call me a terrorist is Ad hominem," he wrote Dec. 15 in a wide-ranging posting.
A military official in Washington said the Army rejected Loughner in 2008 because he failed a drug test. The official spoke on condition of anonymity because privacy laws prevent the military from disclosing such information about an individual's application.
The official did not know what type of drug was detected.
Prosecutors say he scrawled on an envelope the words "my assassination" and "Giffords" sometime before he took a cab to the shopping center. Police said he bought the Glock pistol used in the attack at Sportsman's Warehouse in Tucson in November.
The revelation about the shooter's high-capacity magazines led one longtime Senate gun control advocate, Sen. Frank Lautenberg, D-N.J., to announce plans to re-establish a prohibition that lapsed in 2004 on magazines that feed more than 10 rounds at a time.
At his appearance Monday in a Phoenix courtroom, about 100 miles away from where the shooting took place, Loughner seemed impassive and at one point stood at a lectern as a U.S. marshal stood guard nearby.
His newly appointed lawyer, Judy Clarke, who defended "Unabomber" Ted Kaczynski, stood beside him and whispered to him before the judge ordered him held without bail.
Loughner is charged with one count of attempted assassination of a member of Congress, two counts of killing an employee of the federal government and two counts of attempting to kill a federal employee. Those are federal charges.
State prosecutors, meanwhile, are researching whether they have to wait until after the federal case is resolved, or if they can proceed with local charges at the same time, an official said.
A moment of silence was held Monday evening at the BCS national championship between Oregon and Auburn in Glendale.
Meanwhile, residents of Tucson prepared for memorial services Tuesday for the six who died in the shooting.
The first real community gathering for mourners since the rampage - a Mass for all the victims at St. Odelia's Parish in Tucson - was set for 7 p.m. President Barack Obama was scheduled to arrive in Arizona Wednesday for a memorial service days after calling the attack a tragedy for the entire country.