I've read of soldiers released from service because of their wounds who returned as cooks to their unit because they did not want to leave their friends. I don't know if they were officially reenlisted or just tagged along and were informally accepted by the officers. A lost limb might be too severe, but missing an eye doesn't sound like it would hamper a cook. Hopefully someone has some better information on this.
Junior enlisted personnel who could still fire a weapon and march with their command might be permitted to remain in the ranks, based on what I've found. Others, depending on the nature of their wound, might be able to continue in a support role, such as a teamster or hospital assistant. Actually, when manpower became a grave concern for the Confederates later in the war, an effort was made to send able-bodied clerks and other non-combatants back to the front, to be replaced by those with partial disabilities.
I'm aware of two privates who stayed with their regiments, one who previously had a thumb shot off, and a second who was missing a forefinger. Also recorded was a lieutenant who had lost an eye, and a captain with a partial paralysis in his arm from an earlier battle -they stayed on as line officers. The son of Brigadier General Ambrose Wright still served on his father's staff as an ordnance lieutenant despite the absence of one leg, but he unwisely decided to join a forward reconnaissance group that was surprised by a Federal cavalry squad - he was taken captive because he could not keep up with his comrades.