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?
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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.