How do I use both hard and soft water when I have a water softener installed?
Looking at your second photo, I'd say that the top spigot is soft and the bottom is hard. The bottom is directly connected to the water main, coming up through the concrete, through the pressure regulator (the bell-shaped object) and thus, is connected before the softener.

There is no way to tell from the photos what is coming from the back-yard spigot. Your best bet is to buy a water hardness tester or test kit. I just did a search and currently, you can buy a tester from Amazon for around US$16. If your tap water is hard enough to warrant a softener, it is easy to test if water is direct tap or softened. Try washing your hands with standard hand soap. You should be able to tell the difference after rinsing: if your hands feel slippery, the water is softened. Try this first in the front where you have both so you can tell the difference.

If you decide not to use the softener, there may be a bypass valve on the back of the head of the unit, where the pipes enter.