John -
These are extremely useful bits of information, and I guess almost all that will be gleaned from Bledsoe's CSR cards. Other bits of information exist in the Official Records and elsewhere. The battery was split in early 1863, with Bledsoe in command (of a section?) at Port Hudson (and apparently also Chief of Artillery), and Lieut. R. L. Wood in command of a section (and an ad hoc battery with other units) operating north of Vicksburg. In early May, '63, Bledsoe (and his section??) transfers to Jackson, MS, and then he participates in the Battle of Raymond on 12 May (famously, with a "phantom" Whitworth gun). I haven't yet tracked the battery through the Vicksburg siege, but seems as though Hiram is surrendered at Vicksburg and sent to parole camp, and Lieut. Wood and his guns remain uncaptured. May explain (in part) why Wood is in command a lot after Vicksburg.

During the retreat after his 3 November, 1863, attack on Collierville TN, James R. Chalmers reported: "After Duckworth crossed [the Coldwater] the enemy came up to the river and opened a furious fire of small-arms, which was soon silenced by Captain Bledsoe with a few shots from his rifled gun, by which, it is said, he killed 7 men and 15 horses." Also, Colonel Robert McCullogh (2d Missouri Cavalry) reported: "I may also mention that Captain Bledsoe, with his rifled gun, did good service at Coldwater in repelling the advance of the enemy." The editors of the Official Records presumed this officer to be "H. M." Bledsoe (i.e. Hiram), and now, who knows? Hiram appears to be on leave from late August to the end of December, 1863, but remains (in my mind) an iffy candidate to be this "Captain Bledsoe". I'd love to put him in NW Mississippi, conclusively, or else know where he was at, really.
John L