Your family was a little ahead of mine. The Smiths left Indianola, Iowa on 12 May 1862. They didn't arrive in California until the next year, having spent the winter in Salt Lake City, after being attacked by Shoshone Indians twice in one day, 26 Aug 1862, at the Raft River, Idaho.
The train MAY have stopped when someone was in labor. Many wouldn't travel on Sundays, and many stayed in one location for several days due to illness.
It would be difficult to say what wagon train they were on. You will have to due research if you don't have family records. (We are fortunate enough to have a copy of the Smith's journal).They probably traveled west to what is now Omaha, Nebraska, and picked up the Platte River. The trail was on both sides, with the north side called the "Mormon Trail". This is the one the Smiths used. In western Neb. the trail from Missouri merged with the other trail. Many people kept to the north trail to avoid the "Wild Missourians" The trail heads west into Wyoming,and picks up the Sweetwater River. At that point, there are several "cut-offs" that they may have taken, such as the Sublette Cut-off that my family used. They might have stayed on the main trail and passed through Ft. Bridger. Heading north, they may have detoured through Salt Lake City, or continued on. The trail splits again in Idaho. There was a "Hudspeth Cut-off that some used, although it was rough going. Most passed American Falls.A few miles past that the trail split. If you took the right trail, you were bound for Oregon, to the left, California. It was about thirty miles down this trail that my family was attacked. As you can see, it is impossible to know which way they went without documentation.
Several trails pass the Green River...the main trail, Sublette's Cut-off, and the Lander Road.
In May and June of 2007, we went back to Iowa, and using the journal, followed our family's route out west, following the Oregon Trail, ending near the attack site. We saw many wagon ruts.
I appreciate your comments on the pipeline. I am a retired pipeliner, having worked with Pacific Gas and Elecric's Pipeline Operations for 36 years. The last big job was the M-401, 42" line from Canada.I was involved with the job in my District, about a sixty mile stretch. We had archeologists, Indian Representatives, Fish and Game, ...the list goes on and on. Those people drove us crazy, and must have increased the cost of the project by 25%!
Good Luck, Mark