You can get flannel-like button ups that are made from very thin, breathable cotton materials and layer a dark cotton t-shirt underneath. It’s not perfect but it’s better than a hoodie during the summer months

There are lots of thin flannels, especially in women's sections since fast fashion is usually thinner, lower quality material. Just try to avoid polyester or rayon and look for cotton or similar natural fibers that are less insulating and more sweat wicking.