Did you have actual appraisal done or are you thinking of the Assessed Value as your appraisal? If so, I'm guessing it has Homestead Exemption on it correct? In which case you should be paying re taxes based on the amount minus your homestead. Pool should cause increase eventually. Perhaps I misread the question.
Trish, The value of new construction is added to the existing improvement assessed value.Copies of building permits are also sent to the Assessor. New buildings, additions, and other structures will be added in the assessed value.
I would stop at the county tax assesor office and a better idea.