For short walks I doubt if cotton T's are as inappropriate as some would have you believe.
However, for a proper walk I have a choice of Helly Hansen merino wool long sleeved T, or ex-Army (UK) long johns (I guess that's janes?) and long sleeved vest set. Acquired both from my daughter after she left the Army, I'm unsure if there's a mens / womens fit but they fit me fine.
Merino is popular and resists odour, HH stuff seems to last indefinitely, and works best when it's a snug fit. Synthetics use different fabrics, some more suitable for summer than winter . The Army stuff works well in cooler weather, maybe too hot in the summer for me.