Softeners draw air in intentionally when the brine is exhausted during the recharge. But it should be flushed out during the rinse cycle. If you use water during the regeneration and the internall bypass system is not sealing properly, then some of the air can get out of the tank and thus will not be flushed out until you run your faucets.
The water level is usually about 10″ deep. If you run a faucet near the softener while the salt is being drawn into the system and it tastes salty, then the internal bypass is letting the regeneration water enter the plumbing system.