I'm specifically trying to drill hole saw holes of 2.5" in a 4"x4".
How do I create a hole with a hole saw that is deeper than the tool? (The saw itself)
Are there alternatives to hole saws for thicker material?
You could drill as deep as the hole saw allows, then use a chisel to remove the "plug" left behind. Then drill again with the hole saw. One thing I would highly recommend anytime drilling using a hole saw is to give the chips a place to go. In this scenario I would drill a smaller hole using a spade/forstener bit inside the hole but overlapping with where the hole saw will be cutting. Like the red hole in this diagram:
In my experience this allows the hole saw to work much more efficiently.
Hole saw may not be the best tool for the task. A Forstner bit or something of that sort might be more appropriate. Even a spade bit might be a good choice depending on the material and the diameter of the hole.
Three other options come to mind, none are perfect although # 3 is not bad.