Assuming you’re talking about a small house, say 1,200 square feet and not a tiny house, something well under 1,000 square feet, then a smaller house costs less to build, period. Less materials to build it, fewer man hours, it will cost less than a 2,400 sq. ft. house.
You need to know the zoning and building regulations for your specific building site before you begin the process. Unless you’re willing to accept a design the builder already has, something they build fairly often and have all the plans and approvals for, you may need to begin with an architect or engineer and that adds unpredictable costs. We designed our own home and then had an engineering firm draw up plans and certify them. That was a few thousand dollars.
We’re actually building our home, not hiring a builder. That saves money but costs time. I’m retired, so we’re not losing my working income for me to build the house. We’re using timber from our land, which saves materials costs, but adds costs related to tools and equipment.
As for recommending contractors in a particular area - you have to talk to people that live there and work in the housing industry for guidance. It’s not possible to offer recommendations of any value without intimate local knowledge.