A water softener will strip out some or all of the calcium from the water. If you have a plaster-based finish, it'll need that calcium. So you use the hard water (with calcium) to fill your new pool. But if you keep replacing evaporation with the same hard water, because calcium doesn't evaporate, it'll build up in your pool until you'll have to exchange water to get rid of some of it.
If you replace the the evaporation with soft water, then that will slow or eliminate the calcium build up, and will put off or eliminate the need to replace water because of calcium build up. (Which is what a lot of us do.)
If your house is plumbed like it should be, your indoor plumbing will be soft water. The water coming from the hose bibs outside will be hard water. Which you can confirm with your kit!
Have you loaded up Pool Math on your phone yet? You can create more than one pool in the app. Use one to track the chemistry of your pool. Create a second pool and name it "City Water" and record your results from testing the hard water. Create a third one and name if "Soft Water" and record your results from testing the soft water. That way, over time, you can keep on eye on all three. If you see an increase in the calcium of the Soft Water, then you can troubleshoot if that is coming from the street, or maybe it's time for a new softener.
Otherwise known as the very long answer to your question!
But there's more!! By reading your water meter just before and right after you fill your pool, you can calculate the water volume of your pool. That's the most accurate way to get that number. The pool builder's number is only a rough estimate. Doing math with the pool dimensions will also yield an iffy number. If you can limit the use of water in the house while you're filling, you'll get a very accurate number (turn off all landscape irrigation, limit showers and flushes, postpone laundry until the next day, etc). Knowing your pool's water volume is necessary for dosing the right amount of pool chemicals. The better the number, the better the dosing.