Hi. I think I have a similar situation---the house water goes through a water softner, but the water from the tap outside the house doesn't. I don't know what your outside water looks like, but in the summer, when my turts are outside, I still use the indoor water, mainly because the water that comes from the outside tap looks and smells pretty bad. I doubt if it's harmful, I just don't like the way it looks. (I just run a hose from an indoor spigot in the basement through a window to their stock tanks).
You could test the water from the outside tap for contaminants. If you do use it, I'd let it run for a bit (about half a minute or so) to help clear the pipes, especially if it's not used very often. Do any of the indoor taps have screw on aerators? If so, you might be able to get an adaptor to be able to use a hose indoors (screw off the aerator and attach the adaptor). I do this with
the kitchen sink faucet in the winter months when my RES is indoors.
"You become responsible, forever, for what you have tamed." -Antoine de Saint Exupery-