There are lots of
wholesale t-shirt suppliers that would fit that bill.
SanMar, TSCApparel, AlphaShirt, SSActivewear, etc. Lots of brands have soft t-shirts in that range. You want to look for the word ringspun in the product description Hanes, M&O Knits, Alstyle 1701,etc.
I moved this thread to the wholesale t-shirts forum where you can find even more vendors listed by looking through the threads and FAQ there.