As Harriet's parents, Eliza and George Taylor, had ordinances performed in the Endowment House in Salt Lake City in March of 1864, we know that she and her mother would have been traveling with her father. His promissory note from the Perpetual Emigrating Fund records places him in Florence, Nebraska on 8 August 1863 and the PEF ledger books place him in Utah by at least 15 October 1863.
Further research is needed to identify the name of the company the Taylor family traveled with.
Some sources spell her given name as "Hariett". She died with the married name of McClellan.