Do some real research about the era. The 1860s was the American Civil War, Lincoln's assassination, reconstruction began, the Ku Klux Klan was formed, Andrew Johnson was impeached, Grant was elected president, The First Transcontinental Railroad was completed in 1869, the Suez Canal was opened in 1869, Alfred Nobel created dynamite, the Western Expansion (Go west, young man, go west!!!).... the list goes on. Many people were very devoutly religious, so their local church was also very important to these communities. Celebrations (and more somber events) were always a community happening, especially in the more rural areas, which included most of the country - which had 33 states at the time.
You'll also need to research the everyday lives of the people who lived in the specific era and geographic location where your story takes place. Those living in the cities of the East had very different lives than those who farmed or herded elsewhere; life the South, undergoing post-war reconstruction, was radically different than in the victorious North; Civil War vets from both sides headed west to escape their war memories; and the midwest & west coast were radically different than anything that had come before.
So when you add all of these things together you'll find a "situation" that will suit your story. Here's a few ideas:
Pa brings his daughter into town when he votes. Her future hubby is also in town with his dad for the same reason. They wait outside while their fathers vote and start talking....
Her family is part of a barn raising for her future betrothed (or vice versa)... (It could be a roof raising for the church/town hall/whatever.) Then you could move on into your swimming hole scenario...
He's three or four years older, so she's only about 13 or 14 when he leaves to fight the Civil War (16 year olds could bear arms); when he gets back four years later the little girl is no longer a "little" girl anymore... They meet again at a square dance, or church, or...
The first train comes through town - a major community event - and a huge blast of steam blows her hat off of her head, he chases it down for her....
Her horse spooks and he rescues her...
All of these are, admittedly, also clichés. But sometimes KISS (Keep It Simple, Stupid) is what a script requires.
Again, the issue is the specific era and specific geography. They will both have a tangible effect upon the circumstances of their first meeting.
Hope this has helped....