In 1863, the year the movie is set, the fastest route would have involved a ship to Panama, then a trip across the Panama Railroad, and another ship from Panama to California.

The drawback to this approach for a character in the movie would have been the expense involved. The Panama route was used by middle-to-upper-class individuals willing and able to pay to save time. The last governor of Russian America (and his wife) traveled by this route to reach Alaska, for example.

The route around the southern tip of South America was long, but it was relatively cheap and relatively safe. Traveling overland at this time would have been expensive and somewhat dangerous because of the war. Most cross-continent voyages started in Missouri, which was still a war zone at the time. Many stagecoach companies closed or restricted their operations during the war. Those that remained were able to charge a premium.

In addition, the withdrawal of the U.S. Army from the western plains allowed American Indians to reclaim much of their homeland and otherwise assert themselves militarily. Whites in the western territories and states formed militias that fought the Natives with mixed (and frequently bloody) results.

In short, a trip around the southern tip of South America would have made sense for a lower-class individual from New York City in 1863.