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The drive belt(s) in our 2003 Hyundai Sonata squeal. We had them replaced a few months ago.

Assumed another attempt at adjusting them does not stop the noise, what would be the next thing to do? Replace the battery/ alternator?

Belt squeal can be the result of numerous causes. If you're satisfied that adjustment is correct, it's a case of trying to find which component that's driven by the belt(s) is the cause. As you're probably aware, there are numerous driven components, including alternator, a/c compressor, power steering pump etc.
Diagnosis can be fiddly, but try spraying a small amount of water on each pulley individually to eliminate which one is squealing.
One other thing we have come across on this model is a faulty harmonic balancer/damper pulley. These can cause a squealing noise, particularly on start up. If this turns out to be the problem, you'll need to replace the pulley.