Rotten egg odor in softened hot and cold water

I would sanitize the softener and install a Filox filter ahead of the softener with
a 20 micron spun poly cartridge in a clear housing with a built in bypass.

You want a Vortech tank with a Fleck 2510 SE filter valve on it.

Not from an expert, but since no one else has responded ...

Are you in a drought condition? It could be your well is not pulling in the water it once was or that sulfur has started to contaminate your well. It's interesting that if you use a lot of water, the odor fades only to return later. It could be you have a contaminate in the well that tends to settle and when you pull a lot of water, it temporarily runs the contaminant out.

I would allow the system to sit overnight, then flow enough to clear the pipes and test the water directly from the well and test the water from the outlet of the softner. Run a lot of water until the odor fades and repeat. Your water conditioning people should be able to tell from the results if your resin needs replacing. The second test should tell you if something is building up in your well.

Hydrogen Sulphide (H2S) is a gas found in water.
The gas is less of a problem in a drought situation and made worse when surface conditions are wet.
Running water from the tap for extended periods of time will reduce the concentration of H2S in your H2O.

what is the bleach routine? chlorinate the well?