Should I reveal new areas early or late in the turn?

It's almost always better to trigger new enemies at the very start than at the end:

• Overwatch shots cannot crit and carry an aim penalty against regular shots, except for Snipers with the skill "Opportunist". This pretty much offsets the advantage of Overwatch shots being fired at enemies who are not in cover.
• Overwatch shots have no advantage over regular shots, except for the double Overwatch chance on the Heavy class.
• Overwatch shots are basic shots only - all classes have pretty strong special attacks or double attacks that can only be triggered during your turn.
• If your other soldiers are in position to take an Overwatch shot, they are also able to take a regular shot from the same position, if desired.
• Instead of shooting from the same position, they could first move into a better position and then take their shot.
• They could also retreat completely if necessary.

Given all these disadvantages and almost total lack of advantages, ideally you would never put any of your soldiers into Overwatch except as the very last actions in any given turn, if and only if you are sure that none of your soldiers have triggered new enemies.