You'd be looking for a ringspun cotton or a triblend t-shirt from a high-end brand like American Apparel or Canvas. It all depends on how much you want to pay.
American Apparel and Canvas are on the high end of the market. They each offer both ringspun and triblend styles. You're looking at $3 to $4 per shirt over the cost of a basic cotton tee.
If you want a "value" soft tee, you should consider the Gildan Softstyle or Hanes Nano T. You can get these for only $1 to $2 over the cost of a basic tee, and you'll still feel a noticeable difference.
The AA tri-blends are my go-to shirt. A service like this would be well worth it if they offered either this or the Summer shirt over the normal 2001 / 2102.
American Apparel and Canvas are on the high end of the market. They each offer both ringspun and triblend styles. You're looking at $3 to $4 per shirt over the cost of a basic cotton tee.
If you want a "value" soft tee, you should consider the Gildan Softstyle or Hanes Nano T. You can get these for only $1 to $2 over the cost of a basic tee, and you'll still feel a noticeable difference.