From what I can tell, Aaron's rod is only used in three places while in Egypt.
First when his rod turned into a serpent. (Exodus 7:8-13 NKJV)
Aaron’s Miraculous Rod
8 Then the Lord spoke to Moses and Aaron, saying, 9 “When Pharaoh speaks to you, saying, ‘Show a miracle for yourselves,’ then you shall say to Aaron, ‘Take your rod and cast it before Pharaoh, and let it become a serpent.’ ” 10 So Moses and Aaron went in to Pharaoh, and they did so, just as the Lord commanded. And Aaron cast down his rod before Pharaoh and before his servants, and it became a serpent.
11 But Pharaoh also called the wise men and the sorcerers; so the magicians of Egypt, they also did in like manner with their enchantments. 12 For every man threw down his rod, and they became serpents. But Aaron’s rod swallowed up their rods. 13 And Pharaoh’s heart grew hard, and he did not heed them, as the Lord had said.
Then, the first two plagues:
The First Plague: Waters Become Blood (Exodus 7:19-20 NKJV)
19 Then the Lord spoke to Moses, “Say to Aaron, ‘Take your rod and stretch out your hand over the waters of Egypt, over their streams, over their rivers, over their ponds, and over all their pools of water, that they may become blood. And there shall be blood throughout all the land of Egypt, both in buckets of wood and pitchers of stone.’ ” 20 And Moses and Aaron did so, just as the Lord commanded. So he lifted up the rod and struck the waters that were in the river, in the sight of Pharaoh and in the sight of his servants. And all the waters that were in the river were turned to blood.
The Second Plague: Frogs (Exodus 8:5-6 NKJV)
5 Then the Lord spoke to Moses, “Say to Aaron, ‘Stretch out your hand with your rod over the streams, over the rivers, and over the ponds, and cause frogs to come up on the land of Egypt.’ ” 6 So Aaron stretched out his hand over the waters of Egypt, and the frogs came up and covered the land of Egypt.
Aaron's rod is prominently featured in the transformation into a serpent and the first two plagues. However, as the plagues continue, it seems that God and Moses take a more direct role, and Aaron's rod is no longer mentioned. Is there a logical or symbolic significance to the shift away from Aaron's rod? Can it be inferred why God ceased using Aaron and his rod in the subsequent plagues?