A softener has an air check valve. Its job is to keep air from being sucked in after the brine has been sucked out. If that is not doing its job, the symptom you describe is the result. If you can tell when the softener has regenerated, I expect you will find that it regenerated the night before you get the symptom.
I don't know what it would take to replace the air check valve on your softener. On most softeners, the air check valve is part of the pick-up in the bottom of the brine tank.
Sears parts has parts diagrams etc, so I would see what I could find.
http://www.searspartsdirect.com/model-number/625383500/0582/1090000.html
It may be part of this #26 Water softener brine valve assembly Parts #: 7310202 $ 70.73
(left tip of photo)
http://www.searspartsdirect.com/par...html?modelNumber=625383500&categoryName=Water softener&brandName=KENMORE It would be a shame to replace the whole assembly for a check valve. Maybe soaking that tip area in a several inches of vinegar or stronger deposit remover would fix things. You would know you had fixed it once you could suck on the tube with that pickup tip out of the liquid, and not get air. Sucking and releasing vinegar would help get a flow to bring fresher vinegar to the problem deposits. You would want to wait until the salt was low, and you would interrupt the cycle before the brine tank filled back up with water, I think.
The air is not going to be harmful IMO, so you could live with it for a while too.
