When the dealer shows an Ace, players should adjust their strategy by being more cautious, especially regarding their own hands. It's often advisable to consider taking insurance if they have a strong hand, as the dealer may have a blackjack. Players should also focus on their hand total; for instance, if they have a total of 12-16, they should generally choose to stand, as the dealer is more likely to bust. If they have a lower total, they may need to hit carefully, keeping in mind the risk of busting.