It sounds like she was assessed with the WPPSI, a version of the Weschler IQ assessment for pre-schoolers under age 6, or the WIAT - an achievement test that looks at academic skills like reading and numeracy. I'm more familiar with the WISC, the version used for IQ assessments in older children, so I'm happy to be corrected by someone who knows more about the pre-schooler versions.
If it was the WPPSI, I would talk to the psychologist again, and ask for more explanation of her subtest scores. The subtests provide a lot more information than the overall Full Scale IQ. Frankly, the FSIQ can be pretty meaningless if you have a child with a lot of subtest scatter. It may be that she has very high scores on some subtests and moderate or low scores on others. If so, it will bring her FSIQ under the benchmark for a "gifted" designation, without recognizing her very real abilities and capacities.
The WISC uses outcomes on 4 scales - Verbal Comprehension, Perceptual Reasoning, Working Memory and Processing Speed to assess intelligence. Many gifted programs use the General Ability Index (GAI) which combines the scores of the Verbal Comprehension and Perceptual Reasoning scales. These 2 scales measure the more abstract reasoning skills that correlate with giftedness. The GAI doesn't include the Working Memory and Processing Speed outcomes, that correlate less well with giftedness. These last 2 scales have timing and other issues that may confound an assessment of intelligence. I believe the WPPSI has similar issues.
Once you have a better understanding of the details of the assessment, you'll have a better idea whether to re-assess. You'll also be better prepared for discussions with potential school programs, since you can talk about her abilities as well as her personality characteristics. (I guess here I'd caution against thinking that the test outcomes are the child. By that, I mean expecting that a child with really high spatial relations scores will love math or puzzles, or that a child with low vocabulary scores won't be a strong reader. Don't let the test become a self-fulfilling prophecy - just use it as some useful information that helps understand the whole child.)
Then I would start exploring the different school options in your area. Whether she's identified gifted or not according to any IQ assessment, it's important to find a suitable learning environment that is a good fit for her learning style. That's true for any child.
Look at curriculum guidelines so you can assess where your child is. When you speak with them, ask how they provide for children who have already demonstrated ability in reading, math, etc. No school will admit that they don't do advanced academics. You want details on how they engage a child's interest, how they differentiate for children who know their ABC's or already understand addition and subtraction etc., how they cope with a child with perfectionist tendencies etc.
Absolutely, be happy about your very bright daughter - she sounds wonderful. But it's great that you are involved and caring and want to find the best place for her to learn.
Good luck.