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Physical observables, such as the scattering phase shifts and the bindingenergies, calculated from the non-local HAL QCD potential do not depend on thesink operators used to define the potential. This is called the schemeindependence of the HAL QCD method. In practical applications, the derivativeexpansion of the non-local potential is employed, so that physical observablesmay receive some scheme dependence at given order of the expansion. In thispaper, we compare the $I=2$ $\pi\pi$ scattering phase shifts obtained in thepoint-sink scheme (the standard scheme in the HAL QCD method) and thesmeared-sink scheme (the LapH smearing newly introduced in the HAL QCD method).Although potentials in different schemes have different forms as expected, wefind that, for reasonably small smearing size, the resultant scattering phaseshifts agree with each other if the next-to-leading order (NLO) term is takeninto account. We also find that the HAL QCD potential in the point-sink schemehas negligible NLO term for wide range of energies, which implies a goodconvergence of the derivative expansion in this case, while the potential inthe smeared-sink scheme has non-negligible NLO contribution. Implication ofthis observation to the future studies of resonance channels (such as the $I=0$and $1$ $\pi\pi$ scatterings) with smeared all-to-all propagators is brieflydiscussed. All computations in this paper have been performed at the latticespacing $a\simeq 0.12$ fm ($1/a \simeq 1.6$ GeV) on a $16^3\times 32$ latticewith the pion mass $m_\pi\simeq 870$ MeV.

$I=2$ $\pi\pi$ scattering phase shift from the HAL QCD method with the LapH smearing