If your water smells like rotten eggs, that is hydrogen sulfide. It is a dissolved gas, and it is common in wells across the Highland Rim and the rural edges of the Central Basin — Cannon, DeKalb, Bedford, Coffee, Dickson and Warren counties especially.
The mistake that costs people thousands
A water softener does nothing to hydrogen sulfide.
A softener works by ion exchange: hardness minerals swap onto a resin bed and salt washes them off. It is very good at calcium and magnesium. It is not a filter, and a dissolved gas passes straight through it.
This matters because "my water smells" and "my water is hard" get treated as the same enquiry by a lot of companies. Someone quotes a softener, it gets installed, and the smell is still there on day one. We would rather you heard this before you paid for it than after.
What actually removes it
Hydrogen sulfide has to be oxidised — converted into a solid particle that a filter can then catch. In practice that means an oxidising filter tank ahead of anything else. Media such as Katalox Light handles sulfur along with iron and manganese, which usually turn up together in the same well.
The order matters. Oxidise and filter first, soften second. A softening resin bed fouled with iron is an expensive thing to have to replace.
The question to ask anyone quoting you
Ask them plainly: "which part of this system removes the sulfur?"
If the answer is the softener, or if they cannot name the media, get a second opinion. If they have not tested your well before quoting, they are guessing — iron, manganese, sulfide, pH and hardness all interact, and the right build depends on the actual numbers.
One more thing worth checking
If the smell is only on the hot side, the problem may be your water heater's anode rod reacting with sulfate-reducing bacteria, not the well. That is a much cheaper fix. A company that tells you this is a company worth buying from.