One thing I have to ask first–is there any particular reason you don't want to use your tap water? In many cases, as long as you add a dechlorinator (easy to find, cheap, lasts a long time for small amounts of water) tap water is perfectly fine for common fish like bettas. Otherwise, you're looking at buying a lot of distilled water and remineralizer for weekly water changes.
As a commenter said, I doubt the water source was the problem with your first betta. A new fish tank, even with a filter, will not be safe for fish until bacteria grow in the filter. Why? Because these bacteria will consume the fish's chief waste product, ammonia, and (after a couple more steps) excrete it as nitrate, which is far less toxic. A newly set up tank, without frequently replenishing the water, will rapidly accumulate ammonia which will painfully poison the fish. If you didn't know this—and the fish shop should tell you these things, but many don't know or don't bother–then that is the likely reason the fish didn't make it. Another possibility with bettas is heat; if you didn't heat the tank, it never had a hope for a long life. Bettas are tropical fish that need at least 77 degrees Fahrenheit water temperatures or else become sluggish and very prone to illness.
As for aquarium salt, don't bother. It is an option if the fish gets sick, but there is not salt in these fish's native waters and it adds more risk than potential benefit.
Ideally, you would wait until these aforementioned bacteria have grown in the filter before you add the betta. To describe how to do this in detail is beyond the scope of this question, but the gist of it is that you need to artificially add waste to the tank to feed the bacteria. This process takes patience and ideally some water tests to verify that things are working. See here for some more info. If you don't want to wait for this bacteria growth process to occur (known as "cycling"), then it doesn't matter how long you wait to add the fish...just make sure you change out about 25% of the water every 2-3 days for the first month or two you have it in there.