My opponent used this card with a bayonet charge. I thought my unit was safe staying behind a town hex and out of normal enemy range; not. That card at the right time used with a bayonet charge card cost me the game.
I can't think how an infantry unit could take ground into buildings, make a 2nd combat and also play this card. So I think it is immaterial whether the words 'moves onto' means in Movement Phase or in a general sense of entering the hex during their turn.
Glory is fleeting, but obscurity is forever.