I've just tested mine on the K-3.
Setting the lens to minimum distance then focusing on a distant subject, mine frequently focuses in two stages - one long, and one short (occasionally followed by fine tuning). Sometimes, it takes three stages like yours, and sometimes just one, with fine tuning.
The same thing happens when I set the lens to infinity and focusing on a short-range subject, but it's somewhat less noticeable.
Whether it takes one, two or three stages (followed by fine tuning) seems to depend on the focus target I choose and lighting, which isn't at all surprising.
I hadn't noticed this behaviour previously, but in all honesty I've never watched the distance window when focusing. Also, I typically pre-set the focus ring to infinity or minimum focus distance depending on whether my subject is nearby or distant. I first got into that habit using the HD DA55-300 screw-drive model, where it's pretty much essential if you want reasonably quick AF acquisition. Doing this on the 60-250 significantly reduces the AF time, and I've noticed that it seems less prone to the stepping you've demonstrated.
It's been a couple of months or more since I last used this lens, but taking shots of distant rooftops for this test, I'm blown away by the detail it records, even wide open. From f/5.6 onwards, it's silly sharp
And what beautiful out-of-focus rendering...