In the message, obtained by CBS News and The Associated Press, the Democrat on Friday congratulated Republican Secretary of State Trey Grayson on his new position at Harvard University.
She wrote him: "After you get settled, I would love to talk about what we can do to promote centrism and moderation. I am one of only 12 Dems left in a GOP district (the only woman) and think that we need to figure out how to tone our rhetoric and partisanship down."
On Monday morning, doctors said Giffords was "holding her own" after being shot through the head at an outdoor event to meet constituents two days earlier, but they warned she is still in critical condition and her road to recovery will be long.
Giffords, 40, remained in intensive care at a Tucson hospital. She is still in critical condition and heavily sedated with a breathing tube in place.
Dr. Michael Lemole, the chief of neurosurgery at the University Medical Center told CBS' "The Early Show" co-anchor Erica Hill on Monday that it would be at least another few days to a week before Giffords could hopefully be declared "out of the woods."
A 22-year-old manwith wild beliefs steeped in mistrust faces a federal court hearing Monday on charges he tried to assassinate Giffords and killed others in the Tucson shooting rampage on Saturday.
Late Sunday night, Gifford's husband, astronaut Mark Kelly, said she had been shot while "doing what she loved most - hearing from her constituents."
"On behalf of Gabby and our entire family, I want to extend our heartfelt gratitude to the people of Arizona and this great nation for their unbelievable outpouring of support," said Kelly in the written statement, in which he also expressed his family's condolences for the other victims.
Doctors said Sunday she had responded repeatedly to commands to stick out her two fingers, giving them hope she may survive. Lemole told CBS News that those same tests would continue on Monday.
Lemole said Sunday that the trajectory of the bullet that hit Giffords during the attack on the Democratic lawmaker made them optimistic about her recovery.
The brain surgeon said it was good news that she was able to follow simple commands from the doctors and that the bullet did not cross from one hemisphere of the brain to another. Giffords can't speak yet or open her eyes.
"One thing I'm going to emphasize here is we take those kind of simple commands for granted, but they imply a very high level of functioning in the brain," Lemole said.
Lemole gave an account of how surgeons were able to get Giffords into the operating room within 38 minutes of the attack. He said they were quickly able to control the bleeding, which was not severe or excessive.
"She is still in critical condition, brain swelling at any time can take a turn for the worse," Lemole said.
Lemole said that during the surgery, doctors removed bone fragments caused by the bullet fracture to take pressure off the brain. They also removed some devitalized brain tissue, but, Lemole said, "I'm happy to say we didn't have to do a whole lot of that."
CBS News medical correspondent Dr. Jennifer Ashton described the surgery, called a decompressive craniotomy: "The hair gets shaved. The scalp gets flapped backwards, and a large portion of the cranium skull bone is drilled away. That portion remains in a refrigerator for up to, potentially, two to three months, and (that area of the brain) is left open so the brain can swell and not impair other parts of the brain."
Lemole said Monday morning that he remained "cautiously optimistic," and he cautioned that, while he and other members of the hospital staff will be more certain of her survival in a matter of days, the degree to which she will likely be able to recover from her injuries may take much longer to assess.
"We don't close the book on recovery for years, so it will take as long as it takes," Lemole told "The Early Show's," Hill.