We study the phenomenology of Higgs bosons close to 126 GeV within the scale invariant unconstrained next-to-minimal supersymmetric Standard Model (NMSSM), focusing on the regions of parameter space favoured by low fine-tuning considerations, namely stop masses of order 400 GeV to 1 TeV and an effective $μ$ parameter between 100-200 GeV, with large (but perturbative) $λ$ and low $\tan β=$2-4. We perform scans over the above parameter space, focusing on the observable Higgs cross sections into $γγ$, $WW$, $ZZ$, $bb$, $ττ$ final states, and study the correlations between these observables. We show that the $γγ$ signal strength may be enhanced up to a factor of about two not only due to the effect of singlet-doublet mixing, which occurs more often when the 126 GeV Higgs boson is the next-to-lightest CP-even one, but also due to light stops (and to a lesser extent light chargino and charged Higgs loops). There may be also smaller enhancements in the Higgs decay channels into $WW$, $ZZ$, correlated with the $γγ$ enhancement. However there is no such correlation observed involving the Higgs decay channels into $bb$, $ττ$. The requirement of having perturbative couplings up to the GUT scale favours the interpretation of the 126 GeV Higgs boson as being the second lightest NMSSM CP-even state, which can decay into pairs of lighter neutralinos, CP-even or CP-odd Higgs bosons, leading to characteristic signatures of the NMSSM. In a non-negligible part of the parameter range the increase in the $γγ$ rate is due to the superposition of rates from nearly degenerate Higgs bosons. Resolving these Higgs bosons would rule out the Standard Model, and provide evidence for the NMSSM.