I am NOT an expert on the ACW, so please feel free to believe someone else if they turn up and say "that Wreckedoften guy is full o' crap (and he's not a real Baron, either)".
Based on all the Troiani paintings I've seen that depict them, Union regiments would have the Stars & Stripes on the right, and the regimental flag - usually dark blue, but not always - on the left (as you stand behind them, facing towards the enemy). I believe that any "special" flags over and above these two were carried to the left of the regimental flag.
As far as I am aware, Confederate regiments usually only had the one flag.