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