If you're looking for anime similar to Koneko no Byouki, you might like these titles.
The official music video for Grasshopper Monogatari by Noppo Takami that was featured on NHK's Minna no Uta program.
The official music video for Ringo no Uta by Ringo Sheena that was featured on NHK's Minna no Uta program.