There could be information on his widow's pension application.

Confederate Pension Records
Surname: Reynolds
Given Name: Willliam W.
Application Number: 20381
Widow Surname: Reynolds
Widow Given Name: Sarah N.
Company: F
Regiment: 15 ( Northwest )
State Served From: Arkansas
Division: Infantry
Pension County: Benton
Death Date:
Application Year:
Widow Death Date:
Comments: widow applied 1917 / questionnaire
http://www.ark-ives.com/documenting/confed_pensions/detail/default.aspx?sec=ConfPension&id=21925