In June 1978 we have Rasmussen, using his true identity, quitting his job in Texas to “work elsewhere”. By November of that year, he is in SoCal and in a relationship with Marlyse. We don’t know how long he was there but long enough to become in a relationship. Since he did not have his own child with him at the Thanksgiving gathering and made no mention of being a “ Single Father” to the family, I believe he did not have the mystery child in his custody at that time.
At some point soon after that, they turn up in New Hampshire where he gets a job at Waubec Mills. He was with someone using the name Elizabeth Evans in Feb 1980. That leaves a year for him to move to NH with Marlyse and her children, secure physical custody of his mystery child, murder them all and dispose of their bodies in an area he knew of because he had done work at the nearby home of his boss at Waubec, and then start up a new relationship that had developed to the point that the woman would assume the role of his wife. That is a lot to happen in a year. I think a safe operating assumption would be that Elizabeth Evans was Marlyse.
In Dec 1979, Rasmussen passes a bad check and is arrested later the next year. In 1980, he is arrested twice for stealing electricity, which probably means his was cut off for non-payment and he usued his skill as an electrician to turn it back on. It sounds like he as having serious money problems. If his his job ended, he may have been having problems collecting unemployment insurance or finding an other job due his fake identity. “ Elizabeth” seems to have vanished sometime between May and October 1980. I think it is a safe assumption that the murders occurred during this period.
In the Spring 1981, Rasmussen, still using the name Evans, is photographed with his new girlfriend, the pregnant Denise Beaudin while socializing with neighbors. He was still living at 925 Hayward St Manchester, where he was living with Elizabeth in Feb 1980. He was living with a wife and 3 children, then suddenly they are gone but he remains and later a new girlfriend moves in. Even 40 years later, people remember something like that. They may have moved, they may not follow true crime, but some neighbors remember something about the family and when they left. Publicly will hopefully generate more leads.