Added assessment
An additional assessment imposed when taxable new construction or an improvement is completed after the regular assessment date.
New Jersey example
A completed addition can create an added assessment for the portion of the tax year after completion, so the seller's old bill may not represent the future carrying cost.
Official source
https://www.nj.gov/treasury/taxation/lpt/localtax.shtml
This is a plain-English summary. Rules and dollar amounts change, so check the official source or a professional before you file or sign anything.