I'd call the dealer. First of all, the water shouldn't be tasting salty. Also, your local dealer should be willing to do all the testing for free.
More salt in the tank shouldn't have an affect, but do find out which type of salt works best with your system.

Sounds like you have a problem during the brining stage of generation.

During that stage, the brine in the salt tank is sucked into the resin tank to regenerate the resin. If the brine isn't sucked into the resin tank then the resin isn't regenerated and the service water will not be softened.

The problem may be in the brine pickup, brine pickup air-check valve, float assembly, brine line and it's fittings, or in the control valve (the big thing on top of the resin tank that runs the softener).

Without knowing your water conditions and the brand and/or model of your softener it's difficult to be more specific.

If your softener is still under warranty then call the dealer.

Thanks. I'll call the dealer.

For my own knowledge, if my softener was working properly my water analysis should read calcium, magnesium, and hardness of zero, right? I was told a softener can remove some iron, IRRC ~3 mg/l, so my .11 should also be zero, right?

If your softener is competent, sized correctly, and setup properly for your water conditions and water usage the test results should pretty much zero out and hold that way until it is time to regenerate.

If you see hardness before the softener regenerates then it is too small (in SFR) or not setup properly for your conditions and usage... providing it is a quality water softener to begin with.

Well water conditions can change and should be routinely monitored so the softener can be adjusted to keep it working correctly and... on a well I would be doing routine and regular tests for bacteria.

You are responsible for keeping your water nice AND for keeping it safe!