I like Patagonia tees, but they aren't the only ones around. They wick perspiration, dry quickly and are lighter weight. The are the core of a cold weather layering system. The old rule is "cotton kills" --- fine on a summer afternoon, but a hypothermia bomb shell if conditions go the wrong way.
Other vendors? New Balance, Brooks, REI, Duofold, etc. Most of the stores like Target have wicking tees on one kind or another. Each has its pros and cons-- weight, cost, wicking performance, etc. In general, any of them will be cooler and lighter than a soaked cotton tee at the top of the hill. And much drier and warmer in cold weather.