The "Alledged died in Hospital" comes from the Second Auditor's Office in the Treasury Department in a directive to the Quatermaster General to report what Needham had already been paid and what was still due him. They are not questioning if he is deceased, they are just stating that they are uncertain of the manner/place of his death.
Needham's service card for August-October 1862 reports him as "Absent - sick in hospital". This card covers three months of 1862 - August, September, and October -- which makes it just that much more difficult to pinpoint where the regiment was when Needham got sick and needed to be hospitalized. If he was hospitalized in August 1862 there is a very good chance that the hospital was located in or around Saltillo, MS., as the regiment spent most of July & August 1862 in and around Saltillo, MS. (I do not know if there were any hospitals in Saltillo, MS.). September 1862 was a very busy time for the regiment. In the early part of the month they were in the Waterford, MS., and Lumpkin's Mill, MS., area, they then moved to the Baldwyn, MS., area, and by the middle of the month they were in Iuka, MS. The battle of Iuka took place on September 19 & 20. From September 20th thru the end of October 1862 the regiment was almost constantly on the move -- retreating from Iuka to Baldwyn, joining armies with Van Dorn in Ripley, marching on Corinth, fighting in the battle of Corinth, retreating from Corinth, and by the end of October 1862, they were back in the Waterford/Lumpkin's Mill area.
If I had to guess, I would say that Needham was hospitalized before the regiment started making it's way to Iuka, MS. One reason I would hazard this guess is that he died on October 20, 1862, yet the service card covering that time frame still has him as "sick in hospital". This could indicate that the report of his death took some time to reach his company - which would be the case if they were on the move.
Good Luck!
Kenny