Mr. Wright, as you've already learned, your ancestor's records a not atypical of the soldiers of the 37th Alabama after Vicksburg (or most other Confederate units, for that matter).

I've been researching the 37th for more than 20 years, and my site that's devoted to this study is located at the link below. Here's what I information I've collected on your Sgt Spencer Wright on the the site:

Wright, Spencer
1st Sergeant
Company H
Age at Enlistment: ca 30
WIA or sick as he signed his parole at Vicksburg MS on 10 July 1863 as a Private of Company H of the 37th AL Infantry CSA while hospitalized there; Paid $120.00 on 15 Dec 1863 for May-Oct 1863; Witnessed the Confederate Pension application of Mrs. M.H. Corbett, widow of Henry E Corbett (Co H); Burial record for "Wright, Spencer, Sergeant, Co H, 37th Alabama Infantry, April 30, 1832-Oct. 2, 1903" and a marker for him located in Enterprise City Cemetery, North Main Street, Enterprise AL - photo of gravemarker courtesy of Janice Merrill Johnt in Burials section.

This data is included in the alphabetical roster: <http://www.alabama37th.com/bios-woodh-youn.htm>

I sincerely hope you find this information useful. There are a number of men of the Wright surname in Sgt Wright's Company H who may well be relatives, too. Please let me know if you make any connections, or if I can be of further assistance.

Cordially,
C.C. (Chip) Culpepper in Little Rock, AR (great-grandson of Pvt John M Culpepper of Co B, 37th Alabama Inf)