This usually means brine is not rinsing out of the softener properly at the end of a regeneration.

The softener draws salt water through the resin to recharge it and then flushes it to drain. If that flush is short, blocked, or mistimed, some brine leaves through the house instead.

Look for the taste strongest in the morning after an overnight regeneration, a brine tank holding more water than usual, and a drain line that barely runs during the cycle.

Do not buy a new softener yet, and do not change salt brands. This is usually a setting, an injector, or a drain restriction on the unit already there.