I would not be surprised if Lincoln and Grant were honorary members. The Red Strings were Southerners living in the South during the Civil War but who favored the North. They used a red string to identify themselves to one another. Albion Tourgee was aided by "Union men" while on campaign in the South. He had his protagonist, Col. Comfort Servosse, loosely based on his own life, friends with a Hero of America, Jehu Brown in his novel A Fools Errand.
The Red Strings were a sort of counterpart to the "Copperheads" in the North, except that they formed clubs especially in the mountains of Tennessee.