Last May I took my 2002 Kia Spectra in for the 100,000 mile check up and my mechanic told me that I needed to have the timing belt changed. Up until this point I had not had any problems with the car other than a flat tire. After I picked up the car from the mechanic I started hearing a high-pitched squeal that would happen for 30 seconds to a minute when I’d start the car up in the morning or after it had been sitting for a while. I took the car back and my mechanic told me the timing belt wasn’t on tight enough, fixed it and sent the cxar back. This summer the car only made the squealing a few times, but now that the weather is starting to change the squealing has gotten almost unbearable (loud enough to wake my neighbors when I leave for work at 5:30 in the morning). I have two questions. The first is: Can a bad timing belt change cause this much noise? Second: Should I take the car back to the mechanic a third time or just take it to the dealership?
It’s Time!
Whether or not the timing belt is making the squealing, the timing belt is due to be replaced. This car has an “interference” engine, as do many or most cars with timing belts (Some cars have timing chains). Cars like yours will suffer unnecessary damage in excess of hundreds of dollars if you put off timing belt replacement until it breaks, and it will break. You will be “on foot,” with a damaged engine and a car being towed.
It would be difficult for your mechanic to have tightened your timing belt because of its location, but fairly easy for him/her to have tightened the “serpentine” or accessory belt. The serpentine belt can definitely squeal, especially cold or colder. A timing belt tensioner pulley can be noisy, but I doubt your mechanic got near it (difficult to access).
See if either belt has been replaced. You need both, now. The timing belt replacement is expensive. Make sure you are sitting down when you receive the estimate. If it makes you feel any better, we all go through it and your car will love you for it, like an early Christmas present! By doing the serpentine belt at the same time, there should be no additional labor, only an inexpensive belt cost.
Where it is done is not as big a decision as just getting the car in and doing it, soon. Get an estimate from both places. You decide, as you know the people. The dealer will probably be somewhat higher. Ask if they recommend other parts, while they are “in there” during the surgery. A water pump and timing belt tensioner and tensioner pulley usually make a lot of sense at this time. Ask.
Do it now. The squeal is nothing. A broken timing belt is bad news.
The timing belt is inside the engine and is not squealing. By the way, it needs to be changed on a regular basis (mileage and time… read your owners manual). The squealing is coming from the accessory drive belt(s) outside the engine. You may have one or multiple “vee” belts, or a single “serpentine” belt. These belts had to be removed to get at the timing belt, and may not have been properly tensioned (they should have been replaced, so long as they were off the car anyway). It’s also possible that the belt tensioner (if there is one) was either damaged in the process, or has chosen this moment to go bad. If your mechanic can’t figure it out, try another mechanic or go to the dealer.
Note: I read the OP as that the timing belt was replaced, and then the squealing started. Please clarify which is the case.
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Kia specifies timing belt replacement at 62,500 miles (96,00 km) or 48 months, whichever comes first. If the belt fails - Excuse me…: WHEN the belt fails, say goodbye to your engine. The timing belt has teeth, like a chain, so it cannot slip and squeal. But it does wear out and break without warning. I was not the first person to say goodbye to an engine and you will surely not be the second.
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Earlier, the technician may have tightened your accessory belt or your power steering belt, not your timing belt. A worn accessory belt can certainly can slip and squeal if it is worn out or loose.
Please do not delay replacing the timing belt any longer.
In order to replace the timing belt all the drive belts have to be removed and reinstalled (a good mechanic would replace them too). It’s one of the drive belts that’s making the noise. It’s either too loose or too old or both. Take it back and have them fix it.
Thank you for all of the suggestions 
The squealing did start after I got the timing belt replaced, which is why I thought that’s what might be making the noise. I am thinking I need to take it to the dealership so I can be a good neighbor again.