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The conventional teleportation protocol requires a state entangled in only one degree of freedom, while hyperteleportation requires more than single degree of freedom to complete the task. The hyperteleportation schematics are demonstrated only for the photonic systems, where in the present paper we extend the idea to a hyperteleportation protocol involving the atomic internal and external states. The protocol is deterministically engineered through resonant and off-resonant Atomic Bragg Diffraction (ABD) involving two-level neutral atoms under standard cavity-QED working environment. Moreover, the longer interaction time Bragg's regime with well separated transverse momenta states as an output of the neutral atoms guarantees the high enough engineering fidelities with reduced decoherence rates. The experimental parameters for the demonstration of the proposed scheme are also elucidated briefly describing the optimistic feasibility for the experimental execution of the proposed schematics.

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