To the OP, I'm assuming that your tap water is not ideal for brewing. But what are the actual problems? If it's a question of overmineralization and high alkalinity, then maybe your solution to the overly low alkalinity in that new Walmart water is blending? That is the first place I would seek my alkalinity.
The second option would be to use baking soda, since its readily available. As Jon pointed out, it is important to keep the sodium level modest in most cases. The thing he didn't point out is that if you are adding sparging water to the mash, then the baking soda free sparging water addition will dilute that high sodium content that may exist in the mash. The resulting sodium content in the kettle can be much lower than that required in the mash due to the baking soda use. The supporter's version of Bru'n Water includes this dilution feature, so that you know what your ending sodium content is in the kettle and you can safely add extra baking soda to the mash to meet your pH target. The next supporter's version will be even better. Be aware that the free version is probably never going to be updated. There are good reasons to become a supporter.
By the way, you really have to push the sodium level well above 100 ppm to have any significant taste effects and they only rear their head when you also have a lot of sulfate in the water. Don't be too afraid of sodium. That 50 ppm level is safe.
Yes, Pickling Lime does have to be added directly to the mash or you will cause calcium in your existing water to precipitate out. But in most cases, you are starting out with a water like RO or DI and you don't really need to worry about precipitating calcium out. I still add my lime directly to the mash. PS: All the rest of your mineral and acid additions should be added to the water BEFORE you add the grains. This helps assure that you get all the additions completely mixed and dissolved in the water and EVENLY distributed in your mash. DON'T add minerals to the mash and expect to be able to mix everything well. Matt already proved that it is very difficult to get the additions mixed in the mash.
Gordon's method is good in Dry Stouts and in beers where you DON'T want the roasty flavors in your finished beer. But it is generally a poor substitute for getting the water chemistry correct in the first place and mashing the dark grains in the main mash. In addition, the recipe is likely to need to be modified with much higher dark grain content to account for the poorer color and roast flavor extraction that Gordon's method produces. It is not a panacea. BETTER CHEMISTRY = BETTER BEER