You should soon be getting a “Declaration of New Construction” from the assessor’s office. They will ask about the specific costs to do the addition/remodel (land development costs, building construction costs, site improvements, other costs, financing costs, etc.).
They will also ask about the building characteristics (general, not just the remodel) like number of bedrooms/bathrooms, total square footage, etc.
After you send this in (you have 30 days), they will take a few months to process it and will send you a supplemental bill for the difference.
In our case, the supplemental bill with the new assessment was slightly lower than if they had used the full cost of the addition/remodel, so it looks like they take a combination of your costs and the comps from your neighborhood for similar properties.