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Your detail of digging adjacent to the existing slab can work. You will need to dig back under the slab enough so the new concrete will go under the slab and rest on top of the newly poured pier. Drilling and epoxying rebar into the existing slab and extending it down into the new concrete pier will be good.
You will need to make sure you rent a concrete vibrator to make sure the new concrete freely flows up underneath your existing slab.