N.H. Rev. Stat. Ann. § 658:21-a

If a selectman is absent from or unable to perform his or her duties for all or any part of the day at any state election, the selectman is authorized to appoint a selectman pro tem to perform his or her duties. If one or more selectmen are absent or unable to perform their duties and have not appointed selectmen pro tem, or if a selectman pro tem is absent or unable to perform his or her duties for all or any part of the day, the moderator is authorized to appoint selectmen pro tem to perform their duties.