We don't quite know. The large figure-of-eight tower shields you mention were in use during the Mycenaean period. After the fall of Mycenaean civilsation, Greece entered a period (c. 1100-800 BC) about which we know so little for certain that it has traditionally been referred to as the Dark Age. When we start getting more sources in the eighth century BC, and the Greeks reintroduce writing, we find them already using round shields. These are described in some detail in the Homeric epics.

It's important to stress, though, that these round shields were not the famous hoplite shields of later times. They were flat single-grip shields with shield bosses. The bowl-shaped double-grip aspis that became the standard Greek heavy infantry shield doesn't appear on vase paintings until the last quarter of the 8th century BC, and spreads rapidly throughout the Greek world in the 600s BC.

Other types of shields probably remained in use. Warriors are still depicted using the Dipylon shield and the Boeotian shield all the way down to the end of the Archaic period. Both were oblong shield types with scallops cut out of the sides. It is not known exactly when these shields were completely abandoned in favour of the aspis, but it may have been as late as the Persian Wars.