Stiff, then smooth power steering

Two common causes are a slipping power steering pump drive belt and worn seals inside the rack and pinion assembly. Worn seals will usually affect it differently when turning left or right and it will tend to get better as it warms up.

Be sure the power steering fluid is full too.

Now that I know the history and details, it is possible there is air in the system that hasn't fully bled out yet. Why did you replace the rack and pinion assembly? Was that because of the hard steering or did the hard steering start after the rack replacement?

It's not unheard of for a rebuilt rack to have problems, that's why they have a warranty, but it sure sounds like there is air in the system. Be sure the serpentine belt is tight too. Watch the power steering pump pulley to see if it slows down while a helper turns the steering wheel.

You normally don't have to do anything special except turn fully left and right a few times. Once in a while it can help to do that when parked with the left or right side of the car up higher such as when on a curb.

Look for signs of foaming in the reservoir too. Red power steering fluid will look pink. Clear fluid will look white or cloudy. If the fluid is foamy, the tiny air bubbles are circulating WITH the fluid rather than floating out. Even though those bubbles are small, there's a lot of them and they compress just like a big air pocket. The only cure for that is time. Each time you stop the engine, some of those bubbles will float up in the reservoir and out, but the ones in the rack and hoses will have to circulate their way up to the reservoir the next time the engine is stopped.