I am not much help with your ancestor's military record while outside of Missouri, but I can add a thing or two about what happened to him in the "Show Me" state.
The reason John McLaughlin enlisted in Confederate service far from his Daviess County home was that by August 1862 all of Missouri was under Union occupation. However, during August of that year the Confederacy sent a large recruiting command up from Arkansas to west-central Missouri, and they had considerable success.
The "Lieutenant Davis" you mentioned (also called "Doctor Davis"--so he was probably a doctor or apothecary) was part of the leadership of a small southern guerrilla band that operated for a few weeks in Daviess and Livingston Counties. Perhaps McLaughlin joined LT. Davis in his home area and Davis took him and other southern men south to Ray County where they met a detail from the larger recruiting commmand then operating mostly on the south side of the Missouri River that August. His record may not show it, but it appears your ancestor was a guerrilla at least for a few weeks.
I don't know why McLaughlin remained in the Ray/Clinton County area before he was captured in Clinton County October 15, 1862--according to his prison record. I would have thought he would have traveled south with all the hundreds of other recruits for the long, perilous ride south to Arkansas. Like I said, it was a perilous ordeal, chased by Union cavalry much of the way. Perhaps McLaughlin was separated from his new command, and had to infiltrate back through Yankee patrols back to the Ray/Clinton County area where he had earlier enlisted. People back in Daviess County by then would have known that he had thrown his lot with the Confederacy, so he could not return home or be arrested by the local militia.
I mentioned his prison record. Actually, it is merely a Union clerk's ledger entries about McLaughlin. It says Citizen John McLaughlin of Daviess County was captured or arrested in Clinton County October 15, 1862 and sent to the St. Louis area military prisons. This notation means that the Union military did not recognize McLaughlin's military status, so it is fortunate for him that they did not categorize him as a guerrilla. Such a sentence often carried a death sentence. At St. Louis he was incarcerated in Gratiot Street Prison (prewar the McDowell Medical College) with the notation that he was sent to Washington, D. C. for exchange 2 April 1863. This means that McLaughlin survived a terrible winter in Gratiot Street Prison, so he must have been hearty and/or highly motivated. By the way, I found his brief prison record in Eakin's 1995 book listing the ledger entries for the southern POWs in the St. Louis area prisons.
I hope someone else will be able to assist you with the rest of John McLaughlin's military record and experiences.