  We consider LHC searches for dilepton resonances in an intermediate mass
range, $\sim 10 -80$ GeV. We adopt a kinetically mixed $Z'$ as an example of
weakly coupled new physics that might have evaded detection at previous
experiments but which could still be probed by LHC dilepton spectrum
measurements in this mass range. Based on Monte Carlo simulations, we estimate
that existing data from the 7 and 8 TeV LHC could be used to test values of the
kinetic mixing parameter $\epsilon$ several times smaller than precision
electroweak upper bounds, were an appropriate analysis to be carried out by one
of the experimental collaborations.
