I would say your best bet for the name of the specific boat would be to consult one of the diaries that I previously mentioned, which are at the Tennessee State Library and Archives.

Tenn Archives
Reams Family papers
Jarret Frazier of Voorhies 48 Tn-elected 2nd Lt brevet at Jackson reorg in Sept 62

Frazier says "left prison and arrived by boat near Vicksburg 9/22 pm and anchored–Col Voorhies, Major Gray and other officers came out to see us in skiffs....9/23 put ashore.."

–so apparently the regiment was released as a unit.

Here is another possibility as to your relative, and perhaps how he wound up in Tennessee so quickly after exchange.
This can be found in the Official Records, Series II, Vol 4, p. 436. There are several volumes devoted to "Prisoners of War." And, I believe you can find the names of some of the boats in these volumes.

Sandusky, Ohio, August 26, 1862.
Major W. S. PIERSON,
Commanding Depot of Prisoners of War, Sandusky, Ohio.
MAJOR: The Confederate prisoners of war now at the Sandusky depot will as soon as practicable be transferred via Cairo, Ill., to Vicksburg, Miss., for exchange. They will be prepared to leave at 6 o'clock on Friday morning, 29th, and will be accompanied by a guard of one company. All will be provided with three days' rations....................
Those prisoners who wish to take the oath of allegiance will be permitted to do so and will then be discharged. Duplicate rolls of all so discharged will be prepared and certified to by yourself, one copy to be sent to the Adjutant-General at Washington and the other to the commissary-general of prisoners at Detroit.
Such prisoners will receive any money in your hands belonging to them. Those who belong to Tennessee regiments will be furnished with transportation to Nashville, Tenn., on their giving their parole to report to Governor Johnson.

So there were some from the Tennessee regiments that probabl;y took the oath of allegiance and werer provided transportaion by the Yankees directly to Nashville.

Here is another passage from the same volume and same page:

CAIRO, ILL., August 26, 1862.
L. THOMAS, Adjutant-General U. S. Army:
There is but one gun-boat (the Eastport) here. All others are at or below Helena. I intend leaving on Thursday with all prisoners from Camp Morton (about 3,500) with three transports and the Eastport as convoy to Helena, Ark., and thence to Vicksburg with two gun-boats. It will take eighteen days to return here with gun-boats. I am informed by the general commanding here that he has not the means to receive and to dispose of all the prisoners which may arrive up to that time unless they are placed on transports to await the return of the gun-boats. This would be an expense of $200 each per day. I would respectfully suggest that the remainder of the prisoners not now sent be retained at their respective camps until the return of the transports and gun-boats, eighteen days, then all sent here and down the river at the same time. I await orders.
H. M. LAZELLE,