Early History
Cove was originally called South-Field before it was named. Source
It wasn't until after the Church of Jesus Christ of Latter-day Saints organized a branch in the area. Source page 7.
John Goaslind was the one who named the area, "owing to the geographical location on the east and the foothills and elevation surrounding the community, it being a semi-circle." Source
Mountain Home
Mountain Home was a section of land nescelled on the Utah-Idaho border just north-east of Cove. The people of Mountain Home continued to go to church and school in Franklin, Idaho until the state border was orginized. Mountain Home was discontinued and was consumed by Cove according to federal law.