In the paper we use numerical simulations to show that superlocalization of electronic wave functions takes place on fractal objects also for energies E from the band. Finite size scaling of conductance g versus system size L reveals that  <ln g> scales as L^d_\phi . The values of localization exponent d_\phi we found in 2D are 1.138(3) for the state in the middle of the band E=0.5t, and 1.144(3) for the state near the lower band edge E=-3.5t. These values are in good agreement with the conjecture d_\phi =\zeta _l, where \zeta _l is the chemical length exponent.

PACS numbers: 72.15.Rn, 73.23.--b