If you went to the website you may have misread it, if someone told you that she had to be 6 by October 1st they were misinformed.
Just above the picture of the little boy holding his hand up it states:
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What is the minimum age requirement for kindergarten?
Students are eligible for kindergarten if they turn five years of age on or before October 1 of the year of enrollment. Contact the Gifted and Talented Education office, ###-###-####, for information about early access to kindergarten.
http://www.littletonpublicschools.net/Default.aspx?tabid=105
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This means your child can start kindergarten at 4 years of age if they turn 5 on/or before October 1st.
Your daughter is eligible to start kindergarten this fall when she is a full 5 years old. If she is already 5 years old then perhaps she is in the grade she is supposed to be in and will go to first in the fall when she is a full 6 years old.
Kids are supposed to be:
5 and turn 6 during their kindergarten year
6 turning 7 in 1st grade
7 turning 8 in 2nd grade
8 turning 9 in 3rd grade .........
All the way up to being 17 turning 18 in their senior year of high school.
The web site says she is supposed to be in kindergarten if she was 5 before October 1st. So I would assume since she is in Kindergarten now that she will be in first grade in the Fall. She could have been moved ahead according to this site's information too if the right department had been contacted.
When is her birthday?