I did find a solution for this. A property set of server environment information can be retrieved with the following code:
EnvironmentManager envMgr = new EnvironmentManager();
UID envUID = new UID();
envUID.Value = "{32d4c328-e473-4615-922c-63c108f55e60}";
object envObj = envMgr.GetEnvironment(envUID);
IServerEnvironment3 serverEnvironment = (IServerEnvironment3)envObj;
IPropertySet serverPropsTemp = serverEnvironment.Properties;
There are a number of useful reference paths in the property set that provide the information needed to return absolute URLs instead of relative paths for generated files and other resources.
"AdminTokenURL": "http://REDACTED/arcgis/admin/generateToken",
"RestURL": "http://REDACTED/arcgis/rest",
"AdminUser": "REDACTED",
"SoapURL": "http://REDACTED/arcgis/services",
"AdminURL": "http://REDACTED/arcgis/admin",
"AdminUserPassword": "REDACTED",
"RequestContextURL": "http://REDACTED/arcgis"
Thanks to Ajit for the lead in this ESRI forum thread: http://forums.arcgis.com/threads/62768-Is-it-possible-to-access-HttpContext-inside-SOE