N.H. Rev. Stat. Ann. § 80:1-a

Any town or city may authorize prepayment of the resident tax, and acceptance of the prepayment. In a town, the decision shall be by vote at a town meeting under a proper article in the warrant. In a city, the decision shall be by vote of the governing body. If authorized, any person liable for resident tax may pay the tax at any time from April 1 until he receives notice of the assessed resident tax. The collector shall receive the payment, issue a receipt, and credit the amount paid toward the resident tax later assessed for the year. Each month, the tax collector shall submit a list of all prepayments received to the assessing officials. Once adopted, authority to make and receive prepayments shall continue until rescinded by the same method used for adoption.