It would be helpful to know where he reported for duty after this furlough.

If he reported at Camp Moore, this could be the Skipwith Guards (CO A) of the 27th. LA Regiment. There are several Dunn's listed as members of this unit, possibly his relatives.

From this source:
http://www.acadiansingray.com/27th%20Regt.%20Inf.htm

This regiment was organized at Camp Moore in April, 1862, with 973 men. The regiment left for Vicksburg, Mississippi, on May 1 and arrived on May 4.

The 27th. LA may have moved by rail to Jackson and had a couple of days lay over before leaving for Vicksburg.

Jamie, do you have a copy of the 27th. book?