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We consider a system of $N$ bosons in a unitary box in the grand-canonicalsetting interacting through a potential with scattering length scaling as$N^{-1+\kappa},$ $\kappa\in (0,2/3).$ This regimes interpolate between theGross-Pitaevskii regime ($\kappa=0$) and the thermodynamic limit($\kappa=2/3$). In Forum Math. Sigma 9 (2021), e74, as an intermediate step toprove an upper bound in agreement with the Lee-Huang-Yang formula in thethermodynamic limit, it is obtained a second order upper bound on the groundstate energy for $\kappa<5/9.$ In this paper, thanks to a more careful analysisof the error terms, we extend the above mentioned result to $\kappa<7/12$.

A second order upper bound on the ground state energy of a Bose gas beyond the Gross-Pitaevskii regime