The only way that I know how an independent contractor can compete against the large companies is to find a niche and specialize in doing one thing very, very well.
For example, a general contractor can build houses, commercial projects and large remodeling jobs - and do them well. But if you were to specialize in only doing house restoration, or commercial restoration for buildings damaged by either fire or water - there is a niche that you could compete in that the big boys would have a hard time competing with you. Those small niches can also be very profitable as they have less competition.
Hope this helps!