We learned a TON about this during our sock development and it’s super fascinating… with synthetics, it really is just a matter of quality and price, the best yarns have both geometries as well as chemistry that reduces or eliminates odors. In our socks we use this stuff called Q-Skin which is extruded with silver ions and also has a cross sectional shape that both feels soft and reduces surface area for bacteria to grow/hide. Lots of companies use this stuff including Rapha, Assos, Tracksmith, Lulu Lemon, but it’s like 4-5x the cost of the Asian spin polyamide fibers so it isn’t THAT common. It was a moment though when I was visiting the lab in Italy and it hit me: THIS is why my running shirts from costco stink so bad!!
Meryl Skinlife/Force is another yarn that’s phenomenal at wicking/cooling but won’t retain odors, again, start googling and you’ll find it in socks, shirts, underwear, etc, it’s very similar to Q-Skin.
As for wool, most all wool athletic wear is wool fibers interwoven into a sort of synthetic grid… this was the clever invention of SmartWool years ago in socks where you had this sort of stretchy inner core of elastic/synthetic but the rest of the sock is wool, it’s why over time the sock goes threadbare in some areas exposing the core where the wool is worn away. We use a merino that is spun in Austria in a special process to increase softness and then steam autoclaved which changes the structure of the individual fiber making it much more water resistant and also resistant to bacteria growth. I know that Rapha and Tracksmith use this same yarn producer and I’m sure there are others making shirts from this same stuff.
I will say that I haven’t had the best luck with Smartwool clothing and socks these last 10 years or so, not sure if it’s due to supply chain streamlining after being sold to Timberland and then to VF or just my bad luck, but I have 20 year old Smartwool socks and shirts that are holding up much better than 2 year old ones.
Lastly, my recent go-to wicking travel shirt (I fly ~160-200k miles per year) has been the Patagonia Capilene Cool Merino… not cheap, but ridiculously comfortable, dries fast, never smells, packs smaller than a cotton shirt, and looks new for a lot longer than a standard t-shirt… hard to beat!