It squeals on startup because the alternator is making the most power at that time. The battery has been lowered a bit by the engine start, and the alternator is refilling that. Since the battery was near full, the alternator quickly tapers off to a lower charge rate, and the squealing stops on its own.
However, if you run a lot of accessories (as you will on a rainy day when you have lights, wipers, rear defog and A/C all going to town)... the squealing will return and not let up.
Inspect the equipment. Look for oil contamination on the belt. Look for glazing on the pulleys. Look at belt condition. Fix the oil leak that is causing the contamination - cleaning oil off a belt is prohibitive and replace belt if necessary. Belts are cheap LOL. Wire-brush any glazing off the pulleys - the pulleys that matter are the alternator and crank pulley.
I would say the #1 cause of belt squeal is the belt not being tight enough. Tighten the belt properly - typically more than one might expect, and they don't make it easy - you often have to pry on the alternator in weird ways. A few automakers stick a 3/8" or 1/2" square hole somewhere useful so you can stick a breaker bar on it.
Lastly, don't ignore the possibility that the squeal is a failing bearing in the alternator. They take a tremendous amount of thrust load, which you know all about if you just tightened the belt properly.