I encountered this problem in our last house, as well. Here’s where I will confess that we bought the cheapest, thinnest IKEA rods last time around, cut them with a hacksaw and just butted them up together. If you were to look closely, you’ll see they’re not actually connected and not very pretty. Fortunately, none of our guests ever stood up on their chairs and examined them :)
Photo Credit: Emily A. Clark