Conscripts were frequently assigned to duty in military departments such as Quartermaster, Commissary, Engineer, Medical &c. However, the term "conscript" means a man has been enrolled by operation of the law. Men who were not liable for military service could be hired as civilian employees but these are not conscripts for obvious reasons.
For that matter, the law of Feb. 17, 1864, conscripted free men of color for service in the auxillary departments. The men you have in mind were either soldiers or civilians exempt from conscription working under contract with the government.
You may wish to research inspection reports of military posts in Texas for this type of detail. Officers asigned to the Commissary and Quartermaster Departments were required to list all men serving in their command and the nature of their duties.