The best way to postiviely identify Soloman Alexander's father is through primary records such as birth records, deeds, probate records and such things. The posting that I made was from "The New England Historical and Genealogical Society" publication. In that listing of children of Asa Alexander I do not see a Soloman. But it is possible that there was another Asa living in the same region who did have a son, Soloman.
Your best bet would be to go to a genealogy library or to a Family History Center run by the Mormons and look at records of Winchester for the time that you are interested in. I have no further information about this Alexander family of New Hampshire.