Predicting many molecular properties requires accurate ground-state energies of molecular Hamiltonians. Quantum-selected configuration interaction (QSCI) obtains such energies by measuring an input state to select important electronic configurations and classically diagonalizing the Hamiltonian in the subspace they span. ADAPT-QSCI, an adaptive extension of QSCI, improves the input state through iterative circuit growth. For single-reference molecules, the sampling distribution is dominated by the Hartree-Fock (HF) configuration, so most measurements reproduce a configuration already known. We reduce this redundancy by placing the HF configuration in the diagonalization subspace classically, before any measurement. This invalidates the operator-selection criterion of ADAPT-QSCI, and we derive a consistent replacement with energy and gradient in closed form, adding no query to the quantum device. At the per-iteration shot budgets previously used for ADAPT-QSCI, the proposed scheme reaches chemical accuracy with approximately 63% and 43% of the CNOT gates for H₄ and H₆, and 44% and 40% of the total shots. The advantage depends on the budget, however: in the infinite-shot limit the H₄ CNOT count increases to 125% of that of ADAPT-QSCI. These results identify the per-iteration shot budget as a key design variable governing resource savings in QSCI-based algorithms.