Over the years, many major party nominees have been elected to Congress who started out in electoral politics as members and candidates for a minor party. The newest example is Kyrsten Sinema, who has just been elected to the U.S. House from Arizona as a Democrat. In 2002, she was the Green Party nominee for Arizona State House, 15th district (Tucson). The Green Party listed her in 2002 as one of their star candidates who had a chance of being elected. As a Green, she polled 16.38% of the vote in 2002.
One of the advantages of minor parties in the U.S. is that they often serve as a training ground for aspiring young people who first run as minor party nominees. The experience of running can be educational for the candidate, and may increase self-confidence enough to then switch to a major party.