Steering is hard when cold and belt squells - Hyundai Motor 2003 Elantra

replace your power steering belt they squeal

its the serpintine belt or tentioner. change the belt and the noise will be gone

A common cause of belt noise is a misaligned belt run, this can be caused by worn bearings in the various pulleys causing them to run off track from the pother pulleys, also the crankshaft pulley rubber isolator can come apart and the crank pulley walks outboard and this causes the noise.

Remove and rub hard soap in both sides of the belt and Reinstate.Tide and adjust the belt.

see if you make a a bracket with a adjuster bolt to force the adjuster which is spring loaded up a bit tighter ,yes i know its a bodge but it does work

Make sure your alternator and power steering pumps are turning freely. I personally have never seen a power steering pump lock up, but it could. I have, on the other hand seen numerous alternators freeze up because of bad bearings. Either one would cause a "squalling" sound as the engine runs and tries to rotate the belt over a pulley that isn't turning.

Those cars have manual belt tensioners and sounds like it needs to be adjusted because it has stretched the belt.

Sounds like a loose belt.

There are a few things that can cause a belt to slip like that.

The belt could be not tightened all the way. If it uses a sepentine belt with a tensioner, the tensioner could be bad. If it is not a serpentine belt, then you will have multiple belts and the one that is making the noise can be loose. Use something to pry it tight and tighten down the bolts holding the component th belt goes to.

It can also be a bad pulley. Pulleys have bearings inside of them, at times the bearings will go out and make the pulley hard to turn. You can usually find this easily by taking the belt off and turning the pulley with your hand. If it is hard to turn or makes a grinding noise when turned...it is bad and needs to be replaced.

Old belts can also cause this along with a belt that is wet. There is a spray on Belt Dressing that helps get rid of noisy belts, but I consider that a bandaid to a bigger problem.

You didn't say what type of pump was replaced, water, power steering, or ??? 
If it was the power steering pump you may be low on fluid or the belt is not tight enough. If it was the water pump it still may be the power steering belt was not tighten enough.
Go to www.youtube.com, type in power steering belt and select the appropriate video and it will show you how.

Most likely the belt and pulleys are shiny if you look at them, replace the belt, while you have the belt off try and clean the glaze off the pulley groves, this will make the new belt "grip" better and avoid slipage.