The piping around a water softener can be confusing. A cold water branch before the water softener doesn't necessarily indicate a problem. It's standard procedure to branch off one cold water line to supply outdoor hose connections (and sometimes another to the kitchen sink and refrigerator icemaker if you don't want to drink softened water).
Water for gardens and lawns doesn't need to be softened. The main supply line to the rest of the house continues on to the softener. Once the water leaves the softener, it should branch into two lines: a cold line that runs to interior fixtures and faucets, and a line that feeds the water heater. The outgoing line from the water heater then runs to interior fixtures and faucets that dispense hot water. In addition to those lines, there should be a bypass loop (shown in the photo) with a valve you can open to bypass the softener if you need to disconnect it temporarily without disturbing the water supply to the house.