Please increase the cap on unassigned mods

No. Cull your bad mods and only keep/slice/upgrade good mods. As your quantity of quality mods increases you'll be able to leave more and more characters permanently modded, replacing good mods with great mods and even God mods, trickling the old ones down to B and C teams. Improve your mod stockpile, don't depend on the same handful to be shared by everyone. 
People who make this complaint always have an excess of bad mods. It's one of the few things you don't want to hoard.