Wells, Sump Pumps and Septic Sewage Systems - Rust in water

We have a private well but I don't know the depth of the well at this point. A couple of nights ago when my wife was filling the dogs water bowl the water came out rusty. It started out as a yellowish color, turned a darker rust color and then cleared up after running the water for several minutes. This occurred at a couple of faucets.

However since then the water has had a slight rust color in it. It's not all that noticeable unless you are using a white colored receptacle, such as our dogs water bowl. That was never the case before that night.

Does this indicate some sort of well problem?

I don't know if this is relevant or not but we have a water softener and just a few days earlier I put in 80 pounds of Mortons Rust Remover salt. Could it be as the rust remover salt is working some rust in the pipes is getting knocked loose (for lack of a better phrasing)?

Thank you in advance.

Here's another piece of information. I just did some checking by running water from various faucets in the house and I noticed that the condition is more prevalent if I use just cold water. When I use just the hot water the water comes out clearer than it does on the cold water faucet.

Thank you.