By pass should divert the water away before entering the softener. If your system softens both cold and hot then you may have a leak at the bypass?? perhaps an o ring out of place...or stuck valve partially opened. Try opening and closing fully again.
If your system is plumbed to soften both then you will notice a soft taste in your drinking water.. but the salt taste would also seem to indicate the softener is now working....
The amount of sodium, not "salt" that is added by a water softener that is operating normally/correctly is, 7.85 mg/l (that's roughly a quart) per each gpg (grain per gallon) of ion exchange. I. E. 15 gpg hard water will have 117.75 mg of sodium added to the sodium content in the raw hard water.
That may sound like a lot until you start reading beverage and food labels. A slice of white bread usually has between 120-160 mg od sodium. An 8 oz glass of skim milk, like 500-530 mg. V8 530-560 per glass. Etc. etc. etc.
Usually no one can taste the sodium in softened water. So I say there probably is something wrong with the softener causing incomplete rinse or you have a water leak that allows salty water out into the plumbing when the unit regenerates.
There was a reason why it had not been used lately...
Go to www.kenmorewater.com and their Animations to learn all about how the thing works and how to troubleshoot it to make sure it is working correctly.