Proxima centauri is closer than Alpha centauri, but it is a triple star system.
Then you have Barnard's Star, but it's not visible with the naked eye. Later you'll see Sirius, but it's a double star system.
By the way, there is a strange anecdote about Sirius. There is an African tribe (the Dogons), who seems to have some extraordinary astronomical knowledge. They for instance know that Sirius A has a faint companion Sirius B, but this companion can be observed only with a telescope and was first seen in 1862...
So some claim they were settled on Earth by Aliens thousands of years ago, and that they were coming from Sirius.
So, that left us with Epsilon Eridani.