Mr. Smith,
I am sorry that no one has been able to help you. I dont think that I will be of much help either but here goes: The only JC Lance I found was in a North Carolina unit. There were several Lances in Arkansas and Missouri Units but Joseph C Lance did not appear in any of the rosters I observed. Being close to the Missouri Border, Your ancestor may have served in a Missouri unit. Some of the Missouri State Guard contained Arkansas men, as well as several Missouri regular Confederate units. Many of the State Guard joined Partisan Ranger units like Coleman's and Freeman's (Partisan Ranger) Regiments. Also you may want to check the records for Militia and home guard unitsin the Fulton Baxter County area. Unfortunately a lot of the partisian and home guard records are missing. I think for many of the partisan ranger units, it may be that the records dissapeared on purpose, for self preservation, as many union and confederate officers viewed most of the partisans to be outlaws, and late war and after the war a lot of strife and violence erupted as a result of individuals seeking retribution for treatment of families during the war. You also might check to see if If he had a nickname and possibly he may have signed up under it. Also you might want to check if he had any brothers who were in the Civil war, and check if he served in that unit. also if you know who his friends and neighbors may have been, you may want to see if a Lance is in the same unit and determine if it may be your ancestor.
A few individuals traveled back to their orginial home state to enlist, so you might check the state the family came to Arkansas from. Also there were Union Partisan units that operated primarily in Missouri and Northern Arkansas. Not sure of the legal standing of them. Some of the volunteer units that were never raised to a battalion or regimental level were not always formally recognized by the US Army. I have a friend whose ancestor was not recognized for Federal service in a unit organized in the Batesville, Arkansas area, because the unit was never officially mustered into active Federal service, and the regiment never reached full strength and was disbanded. I think many of the Confederate Milita and home guard units were never offically recognized as Confederate soldiers. So finding him may be a challenge. I hope this helps you.