Using perturbative expansions and the contractor renormalization (CORE) algorithm, we obtain effective hard-core bosonic Hamiltonians describing the low-energy physics of S=1/2 spin-dimer antiferromagnets known to display supersolid phases under an applied magnetic field. The resulting effective models are investigated by means of mean-field analysis and quantum Monte Carlo simulations. A "leapfrog mechanism," through means of which extra singlets delocalize in a checkerboard-solid environment via correlated hoppings, is unveiled that accounts for the supersolid behavior.