The reliability of quantum computation critically depends on the performance of quantum error-correcting codes (QECCs). Performance of QECCs can be severely degraded by hook errors, which effectively reduce the code distance. We develop a systematic framework for bare-ancilla syndrome extraction in $[[n,k,3]]$ stabilizer codes. It provides a criterion for a specific single ancilla fault model and jointly searches for suitable stabilizer generators and data-ancilla interaction orders to correct hook errors. We call such stabilizer generator sets bare-ancilla codes (BACs). We also present two constructions that enlarge the set of valid BACs while preserving minimum distance, either by keeping the number of logical qubits fixed or by increasing it. Under the correlated-data error model, $3177$ $[[7,1,3]]$ graph codes are found to be BACs out of $3379$ and all $335415$ $[[8,1,3]]$ graph codes are BACs. We tested $500$ $[[7,1,3]]$ and $1000$ $[[8,1,3]]$ graph codes in the presence of anisotropic and depolarizing noise. The number of data-ancilla interactions is an important predictor of the pseudo-threshold of these graph codes. In the analyzed codes, the BAC, which works with the bare ancilla method, usually performs well or better than the flag method, particularly for depolarizing noise. Notably, we report new bare ancilla codes, namely $[[6,1,3]]$ and $[[7,1,3]]$ with improved code rate compared to the bare code used in the work of Muyuan Li \emph{et al.} and Maheshwari \emph{et al.} respectively.

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