It may be a tiny bit late for drastic changes, but the idea just came to me now and it might find consideration later, if time permits.
The hidden movement phase has two functions: 1) It shows what the soldiers perceive during the aliens' turn, so that alien's the player can retrace aliens' positions from the directions of bullets and civilians. 2) the waiting creates tension.
For me, the waiting also creates aggression towards the game developers. I don't like to be made to wait. I'd rather the soldiers did the thinking themselves. If they were shot at from some direction, they might just mark an area as potential alien ground. If they saw an alien pass them by, without noticing them, they might indicate, that an alien of type X went "that way". In game terms they could do so by overlaying the map (including the fog of war) with an arrow. This would also give the game more of a role playing feel. Additionally, the game could replay the soldiers memory of what they saw during the hidden movement phase on demand. It shouldn't need to be accurate every time though, as memory never is. If soldiers return fire, the game would need to show. But if they are killed or unconscious, it shouldn't, as they wouldn't live to tell.