  A new approach to exploring low-temperature excitations in finite-dimensional
lattice spin glasses is proposed. By focusing on bond-diluted lattices just
above the percolation threshold, large system sizes $L$ can be obtained which
lead to enhanced scaling regimes and more accurate exponents. Furthermore, this
method in principle remains practical for any dimension, yielding exponents
that so far have been elusive. This approach is demonstrated by determining the
stiffness exponent for dimensions $d=3$, $d=6$ (the upper critical dimension),
and $d=7$. Key is the application of an exact reduction algorithm, which
eliminates a large fraction of spins, so that the reduced lattices never exceed
$\sim10^3$ variables for sizes as large as L=30 in $d=3$, L=9 in $d=6$, or L=8
in $d=7$. Finite size scaling analysis gives $y_3=0.24(1)$ for $d=3$,
significantly improving on previous work. The results for $d=6$ and $d=7$,
$y_6=1.1(1)$ and $y_7=1.24(5)$, are entirely new and are compared with
mean-field predictions made for d>=6.
