One thing to speed up the game some would be to remember the aim per target. Most times the auto-aim is not useful, so the player needs to manually aim, but this is for each shot. Firing a 1AP weapon 4 times like this is time consuming. It would take less time if we only had to aim the first shot, and then just press fire (a new button: auto/manual/again) for any subsequent shots at the same target.
Even better would be to leave the animation in the "aimed" state so the the shooter doesn't need to: step out of cover, aim, shoot, step back into cover X 4. Instead: step out, aim, shoot, shoot, shoot, shoot, step back to cover (by using the "again" button), or step out, aim, shoot, shoot, "manual" (select different target).