  In the pursuit of higher critical temperature of superconductivity,
quasi-flat electronic bands and van Hove singularities in two dimensions (2D)
have emerged as a potential approach to enhance Cooper pairing on the basis of
mean-field expectations. However, these special electronic features suppress
the superfluid stiffness and, hence, the Berezinskii-Kosterlitz-Thouless (BKT)
transition in 2D superconducting systems, leading to the emergence of a
significant pseudogap regime due to superconducting fluctuations. In the
strong-coupling regime, one finds that superfluid stiffness is inversely
proportional to the superconducting gap, which is the predominant factor
contributing to the strong suppression of superfluid stiffness. Here we reveal
that the aforementioned limitation is avoided in a 2D superconducting
electronic system with a quasi-flat electronic band with a strong pairing
strength coupled to a deep band with weak electronic pairing strength. Owing to
the multiband effects, we demonstrate a screening-like mechanism that
circumvents the suppression of the superfluid stiffness. We report the optimal
conditions for achieving a large enhancement of the BKT transition temperature
and a substantial shrinking of the pseudogap regime by tuning the intraband
couplings and the pair-exchange coupling between the two band-condensates.
